Homological Algebra of Noncommutative 'Spaces' I

Introduction

A considerable part of this manuscript is based on the notes of a lecture course in noncommutative algebraic geometry given at Kansas State University during the Fall of 2005 and Spring of 2006 and on (the second half of) my lectures delivered at the School on Algebraic K-Theory and Applications which took place at the International Center for Theoretical Physics (ICTP) in Trieste during the last two weeks of May of 2007 (see [R2], or [R3]). The starting point of the actual lectures in Kansas was the homological algebra of exact categories as it is viewed by Keller and Vossieck [KeV]. Besides an optimization of the Quillen’s definition of an exact category, they observed that the stable categories of exact categories with enough injectives have a suspension functor and triangles whose properties give a ’one-sided’ version of Verdier’s triangulated category, which they call a suspended category. A short account on this subject is given in Appendix K. The main body of the text reflects attempts to find natural frameworks for fundamental homological theories which appear in noncommutative algebraic geometry. The first move in this direction is the replacing exact categories with a much wider class of right exact categories. These are categories endowed with a Grothendieck pretopology whose covers are strict epimorphisms. The dual structures, left exact categories, appear naturally and play a crucial role in a version of K-theory sketched in Section 7 of this work. Sections 1 and 2 contain generalities on right exact categories. In Section 1, we introduce right exact (not necessarily additive) categories and sketch their basic properties. We define Karoubian right exact categories and prove the existence (under certain conditions) of the Karoubian envelope of a right exact category. We observe that any k-linear right exact category is canonically realized as a subcategory of an exact k-linear category – its exact envelope. In Section 2, we consider right exact categories with initial objects. The existence of initial (resp. final) objects allows to introduce the notion of the kernel (resp. cokernel) of a morphism. Most of the section is devoted to some elementary properties of the kernels of morphisms, which are well known in the abelian case. Section 3 is dedicated to satellites on right exact categories. Its content might be regarded as a non-abelian and non-additive (that is not necessarily abelian or additive) version of the classical theory of derived functors. We introduce ∂∗-functors and prove the existence of the universal ∂∗-functors on a given right exact ’space’ with values in categories with kernels of morphisms and limits of filtered diagrams. We establish the existence of a universal ’exact’ ∂∗-functor on a given right exact ’space’. The latter subject naturally leads to a general notion of the costable category of a right exact category, which appears in Section 4. We obtain (by turning properties of costable categories into axioms) the notion of a (not necessarily additive) cosuspended category. We introduce the notion of a homological functor on a cosuspended category and prove the existence of a universal homological functor.

New Prairie Press, 2014 In Section 5, we introduce projective objects of right exact categories (and injective objects of left exact categories). They play approximately the same role as in the classical case: every universal ∂∗-functor annihilates pointable projectives (we call this way projectives which have morphisms to initial objects); and if the right exact category has enough projectives, then every ’exact’ ∂∗-functor which annihilates all projectives is universal. Starting from Section 6, a (noncommutative) geometric flavor becomes a part of the picture: we interpret svelte right exact categories as dual objects to (noncommutative) right exact ’spaces’ and ’exact’ functors between them as inverse image functors of morphisms of ’spaces’. We introduce a natural left exact structure on the category of right exact ’spaces’. Inverse image functors of its inflations are certain localizations functors. Section 7 is dedicated to the first applications: the universal K-theory of right exact ’spaces’. We define the functor K0 and then introduce higher K-functors as satellites of ∗ K0. More precisely, the K-functor appears as a universal contravariant ∂ -functor on a left exact category over the left exact category of right exact ’spaces’. Here ‘over’ means an ’exact’ functor to the category left exact ’spaces’. Our K-functor has exactness properties which are expessed by the long ’exact’ sequences corresponding to those ’exact’ localizations which are inflations. In the abelian case, every exact localization is an inflation. Quillen’s localization theorem states that the restriction of his K-functor to abelian categories has a natural structure of a an ’exact’ ∂-functor. It follows from the universality of the K-theory defined here, that there exists a unique morphism from the Quillen’s K- Q a functor K• to the universal K-functor K• defined on the left exact category of ’spaces’ represented by abelian categories. In Section 8, we introduce infinitesimal ’spaces’. We establish some general facts about satellites and then, as an application, obtain the devissage theorem in K-theory. It is worth to mention that infinitesimal ’spaces’ is a serious issue in noncommutative (and commutative) geometry: they serve as a base of a noncommutative version of Grothendieck-Berthelot crystalline theory and are of big importance for the D-module theory on noncommutative ’spaces’. We make here only a very little use of them leaving a more ample development to consequent papers. The remaining five sections appear under the general title “complementary facts”. In Section C1 (which complements Section 3), we look at some examples, which acquire importance somewhere in the text. In Section C2, we pay tribute to standard techniques of homological algebra by expanding the most popular facts on diagram chasing to right exact categories. They appear here mainly as a curiosity and are used only once in the main body of the manuscript. Section C3 is dedicated to localizations of exact and (co)suspended categories. In particular, t-structures of (co)suspended categories appear on the scene. Again, a work by Keller and Vossieck, [KV1], suggested the notions. Section C4 is dedicated to cohomological functors on suspended categories and can be regarded as a natural next step after the works [KeV] and [Ke1]. It is heavily relied on Appendix K, where the basic facts on exact and suspended categories are gathered, following the approach of B. Keller and D. Vossieck [KeV], [KV1], [Ke2], except for some complements and most of proofs, which are made more relevant to the rest of the work. We consider cohomological functors on suspended categories with values in exact categories and prove the existence of a universal cohomological functor. The construction of the universal functor gives, among other consequences, an equivalence between the bicategory of Karoubian suspended svelte

2 categories with triangle functors as 1-morphisms and the bicategory of exact svelte Z+- categories with enough injectives whose 1-morphisms are ’exact’ functors. We show that if the suspended category is triangulated, then the universal cohomological functor takes values in an abelian category, and our construction recovers the abelianization of triangulated categories by Verdier [Ve2]. It is also observed that the triangulation of suspended categories induces an abelianization of the corresponding exact Z+-categories. We conclude with a discussion of homological dimension and resolutions of suspended categories and exact categories with enough injectives. These resolutions suggest that the ’right’ objects to consider from the very beginning are exact (resp. abelian) and (co)suspended (resp. n triangulated) Z+-categories. All the previously discussed facts (including the content of Appendix K) extend easily to this setting. In Section C5, we deﬁne the weak costable category of a right exact category as the localization of the right exact category at a certain class of arrows related with its projectives. If the right exact category in question is exact, then its costable category is isomorphic to the costable category in the conventional sense (reminded in Appendix K). If a right exact category has enough pointable projectives (in which case all its projectives are pointable), then its weak costable category is naturally equivalent to the costable category of this right exact category deﬁned in Section 4. We study right exact categories of modules over monads and associated stable and costable categories. The general constructions acquire here a concrete shape. We introduce the notion of a Frobenius monad. The category of modules over a Frobenius monad is a Frobe- nius category, hence its stable category is triangulated. We consider the case of modules over an augmented monad which includes as special cases most of standard homological algebra based on complexes and their homotopy and derived categories.

A large part of this manuscript was written during my visiting Max Planck Institut f¨urMathematik in Bonn and IHES. I would like to express my gratitude for hospitality and excellent working conditions.

3 1. Right exact categories.

1.1. Right exact categories and (right) ’exact’ functors. We define a right exact category as a pair (CX , EX ), where CX is a category and EX is a pretopology on CX whose covers are strict epimorphisms; that is for any element M −→ L of E (– a cover), −→ the diagram M ×L M −→ M −→ L is exact. This requirement means precisely that the pretopology EX is subcanonical; i.e. every representable presheaf is a sheaf. We call the elements of EX deflations and assume that all isomorphisms are deflations. 1.1.1. The coarsest and the finest right exact structures. The coarsest right exact structure on a category CX is the discrete pretopology: the class of deflations coincides with the class Iso(CX ) of all isomorphisms of the category CX . s Let EX denote the class of all universally strict epimorphisms of CX ; i.e. elements of s e f EX are strict epimorphisms M −→ N such that for any morphism Ne −→ N, there exists a cartesian square f M −−−→e M f    e y cart y e f Ne −−−→ N s whose left vertical arrow is a strict epimorphism. It follows that EX is the finest right exact structure on the category CX . We call this structure canonical. s If CX is an abelian category or a topos, then EX consists of all epimorphisms. s If CX is a quasi-abelian category, then EX consists of all strict epimorphisms.

1.1.2. Right ’exact’ and ’exact’ functors. Let (CX , EX ) and (CY , EY ) be right F exact categories. A functor CX −→ CY will be called right ’exact’ (resp. ’exact’) if it e maps deflations to deflations and for any deflation M −→ N of EX and any morphism f Ne −→ N, the canonical arrow

F (Ne ×N M) −−−→ F (Ne) ×F (N) F (M) is a deflation (resp. an isomorphism). Thus, the functor F is ’exact’ if it maps deflations to deflations and preserves pull-backs of deflations. F 1.1.3. Weakly right ’exact’ and weakly ’exact’ functors. A functor CX −→ CY is called weakly right ’exact’ (resp. weakly ’exact’) if it maps deflations to deflations and for any arrow M −→ N of EX , the canonical morphism

F (M ×N M) −→ F (M) ×F (N) F (M) is a deﬂation (resp. an isomorphism). In particular, weakly ’exact’ functors are weakly right ’exact’. 1.1.4. Note. Of cause, ’exact’ (resp. right ’exact’) functors are weakly ’exact’ (resp. weakly right ’exact’). In the additive (actually, a more general) case, weakly ’exact’ functors are ’exact’ (see 2.5 and 2.5.2).

4 1.2. Proposition. (a) Let (CX , EX ) be a svelte right exact category. The Yoneda ∗ jX embedding induces an ’exact’ fully faithful functor (C , E ) −−−→ (C , Es ), where X X XE XE C is the category of sheaves of sets on the presite (C , E ) and Es the family of all XE X X XE universally strict epimorphisms of CXE (– the canonical structure of a right exact category). ϕ∗ (b) Let (CX , EX ) and (CY , EY ) be right exact categories and (CX , EX ) −−−→ (CY , EY ) ∗ ϕe a right weakly ’exact’ functor. There exists a functor CXE −−−→ CYE such that the diagram

ϕ∗ C −−−→ C X Y ∗   ∗ jX y y jY ∗ ϕe CXE −−−→ CYE

∗ ∗ ∗ ∗ ∗ quasi commutes, i.e. ϕe jX ' jY ϕ . The functor ϕe is deﬁned uniquely up to isomorphism and has a right adjoint, ϕe∗. Proof. (a) The argument is the same as the part (i) of the proof of K5.2. (b) The argument coincides with the proof of K5.4. 1.3. Interpretation: ’spaces’ represented by right exact categories. Right weakly ’exact’ functors will be interpreted as inverse image functors of morphisms between w ’spaces’ represented by right exact categories. We consider the category Espr whose objects are pairs (X, EX ), where (CX , EX ) is a svelte right exact category. A morphism ϕ from (X, EX ) to (Y, EY ) is a morphism of ’spaces’ X −→ Y whose inverse image functor ϕ∗ CY −→ CX is a right weakly ’exact’ functor from (CY , EY ) to (CX , EX ). The map which o assigns to every ’space’ X the pair (X, Iso(CX )) is a full embedding of the category |Cat| w of ’spaces’ into the category Espr . This full embedding is a right adjoint functor to the forgetful functor w o Espr −−−→ |Cat| , (X, EX ) 7−→ X.

1.4. Proposition. Let (CX , EX ) and (CY , EY ) be additive right exact categories and F CX −→ CY an additive functor. Then (a) The functor F is right weakly ’exact’ iff it maps deflations to deflations and the sequence F (e) F (Ker(e)) −−−→ F (M) −−−→ F (N) −−−→ 0 is exact for any deflation M −→e N. (b) The functor F is weakly ’exact’ iff it maps deflations to deflations and the sequence

F (e) 0 −−−→ F (Ker(e)) −−−→ F (M) −−−→ F (N) −−−→ 0 is ’exact’ for any deﬂation M −→e N.

5 Proof. (a) Notice that each arrow of EX has a kernel, because the square

Ker(e) −−−→ M     y cart y e 0 −−−→ N is cartesian, and it exists when e ∈ EX . This observation allows to use the argument of K4.3 which proves the assertion. e (b) If the category CX is additive and there exists the kernel of M −→ N, then M ×N M is canonically isomorphic to the coproduct of M and Ker(e). In fact, we have a commutative diagram

∼ −−−→ Ker(p2) −−−→ Ker(e)  0    k2 y y j (1) p1 ∆M −−−→ e M −−−→ M ×N M −−−→ M −−−→ N p2 (borrowed from the argument of K4.3). Its left vertical arrow and the diagonal morphism ∆ M ∼ M −−−→ M ×N M determine an isomorphism M ⊕ Ker(p2) −→ M ×N M, which implies ∼ an isomorphism M ⊕ Ker(e) −→ M ×N M. Since the functor F is additive, we have canonical isomorphisms F (M ×N M) ' F (M ⊕ Ker(e)) ' F (M) ⊕ F (Ker(e)). There is a commutative diagram

α F (M ×N M) −−−→ F (M) ×F (N) F (M)     oy yo (2) id⊕β F (M) ⊕ F (Ker(e)) −−−→ F (M) ⊕ Ker(F (e))

α β in which F (M ×N M) −−−→ F (M) ×F (N) F (M) and F (Ker(e)) −−−→ Ker(F (e)) are natural morphisms. Since the vertical arrows of (2) are isomorphisms, this shows that α is an isomorphism iﬀ β is an isomorphism. This and (a) imply that the functor F is weakly ’exact’ iﬀ the sequence

F (e) 0 −−−→ F (Ker(e)) −−−→ F (M) −−−→ F (N) −−−→ 0 is exact for every deﬂation M −→e N. 1.5. Karoubian envelopes of categories and right exact categories. p 1.5.1. Lemma. Let M be an object of a category CX and M −→ M an idempotent (i.e. p2 = p). The following conditions are equivalent: (a) The idempotent p splits, i.e. p is the composition of morphisms M −→e N −→j M such that e ◦ j = idN .

6 id M (b) There exists a cokernel of the pair M −→M. −→p

id M (c) There exists a kernel of the pair M −→M. −→p

If the equivalent conditions above hold, then Ker(idM , p) ' Coker(idM , p).

p Proof. (b) ⇐ (a) ⇒ (c). If the idempotent M −→ M is the composition of M −→e N id j e −→M and N −→ M such that e ◦ j = idN , then M −→ N is a cokernel of the pair M M, −→p because e ◦ p = e ◦ j ◦ e = e and if M −→t L any morphism such that t ◦ p = t, then t = (t ◦ j) ◦ e. Since e is an epimorphism, there is only one morphism g such that t = g ◦ e. This shows that (a) ⇒ (b). The implication (a) ⇒ (c) follows by duality.

id M (b) ⇒ (a). Let M −→e N be a cokernel of the pair M −→M. Since p ◦ p = p, there −→p j exists a unique morphism N −→ M such that p = j ◦ e. Since e ◦ j ◦ e = e ◦ p = e = idN ◦ e and e is an epimorphism, e ◦ j = idN . The implication (c) ⇒ (a) follows by duality.

1.5.2. Deﬁnition. A category CX is called Karoubian if each idempotent in CX splits. It follows from 1.5.1 that CX is a Karoubian category iﬀ for every idempotent p M −→ M in CX , there exists a kernel (equivalently, a cokernel) of the pair (idM , p).

1.5.3. Proposition. For any category CX , there exists a Karoubian category CXK ∗ kX and a fully faithful functor CX −→ CXK such that any functor from CX to a Karoubian ∗ category factors uniquely up to a natural isomorphism through kX . Every object of CXK is ∗ a retract of an object kX (M) for some M ∈ ObCX .

Proof. Objects of the category CXK are pairs (M, p), where M is an object of the p 2 category CX and M −→ M is an idempotent endomorphism, i.e. p = p. Morphisms f (M, p) −→ (M 0, p0) are morphisms M −→ M 0 such that fp = f = p0f. The composition f g gf of (M, p) −→ (M 0, p0) and (M 0, p0) −→ (M 00, p00) is (M, p) −→ (M 00, p00). It follows from p q this deﬁnition that (M, p) −→ (M, p) is the identical morphism. If (M, p) −→ (M, p) is an q q idempotent, then it splits into the composition of (M, p) −→ (M, q) and (M, q) −→ (M, p). q q q The composition of (M, q) −→ (M, p) −→ (M, q) is (M, q) −→ (M, q), which is the ∗ kX identical morphism. The functor CX −→ CXK assigns to each object M of CX the pair g g (M, idM ) and to each morphism M −→ N the morphism (M, idM ) −→ (N, idN ).

F FK For any functor CX −→ CZ to a Karoubian category CZ , let CXK −→ CZ denote a functor which assigns to every object (M, p) of the category CXK the kernel of the pair ∗ F (idF (M),F (p)). It follows that FK ◦ KX ' F . In particular, for any functor CX −→ CY ,

7 FK there exists a natural functor CXK −→ CXK such that the diagram

F C −−−→ C X Y ∗   ∗ KX y y KY FK CXK −−−→ CYK quasi-commutes. The map F 7−→ FK deﬁnes a (pseudo) functor from Cat to the category KCat of Karoubian categories which is a left adjoint to the inclusion functor. This implies, in particular, the universal property of the correspondence CX 7−→ CXK . p For every object (M, p) of the category CXK , the morphism (M, p) −→ (M, idM ) ∗ splits; i.e. (M, p) is a retract of kX (M) = (M, idM ).

The category CXK in 1.5.3 is called the Karoubian envelope of the category CX . 1.5.4. Karoubian envelopes of right exact categories.

1.5.4.1. Deﬁnition. We call a right exact category (CX , EX ) Karoubian if the category CX is Karoubian and any split epimorphism of the category CX is a deﬂation.

1.5.4.2. Proposition. Let (CX , EX ) be a right exact category. Suppose that for p f every idempotent M −→ M in CX and every morphism N −→ M such that f = pf, there exists a cartesian square f 0 N 0 −−−→ M   0   e y cart y p f N −−−→ M

Then the Karoubian envelope CXK of CX has a structure EXK of a right exact Karoubian ∗ kX category such that the canonical functor CX −→ CXK is an ’exact’ functor from (CX , EX ) to (CXK , EXK ). The right exact Karoubian category (CXK , EXK ) is universal in the following sense: every (weakly) right exact functor from the right exact category (CX , EX ) to a right exact Karoubian category (CY , EY ) is uniquely represented as the composition of the canonical exact, hence ’exact’, functor from (CX , EX ) to its Karoubian envelope

(CXK , EXK ) and a (weakly) right exact furnctor from (CXK EXK ) to (CY , EY ). e Proof. (a) Let CX be a category and M −→ L a split epimorphism; i.e. there exists j g a morphism L −→ M such that e ◦ j = idL. Let N −→ L be a morphism. Since j is a g jg monomorphism, a pullback of N −→ L ←−e M exists iﬀ a pullback of N −→ M ←−je M. exists and they are isomorphic to each other. Notice that p = je is an idempotent and a f morphism N −→ M factors through L −→j M iﬀ f = pf. Thus, we have cartesian squares

f 0 f 0 N 0 −−−→ MN 0 −−−→ M     0   0   e y cart y p and e y cart y e f g N −−−→ MN −−−→ L

8 It follows from the right cartesian square that the morphism e0 is a split epimorphism, because it is a pullback of a split epimorphism. (b) Suppose that the condition of 1.5.4.2 holds, and consider a pair of morphisms f q (N, u) −→ (M, q) ←− (M, p) of the Karoubian envelope CXK . By deﬁnition, fu = qf = f and qp = pq = q. By the hypothesis, there exists a pullback N ×f,q M. The equality q0 qf = f implies that the projection N ×f,q M −→ N splits, i.e. there exists a morphism 0 j 0 0 0 0 0 N −→ N ×f,q M such that q j = idN . Set u = j q . Then f 0 0 (N ×f,q M, u ) −−−→ (M, p)   0   q y y q f (N, u) −−−→ (M, q) is a cartesian square in CXK . This shows that split epimorphisms of CXK are stable under base change. The class of deﬂations EXK consists of all possible compositions of morphisms ∗ of kX (EX ) and split epimorphisms. F (c) By the universal property of Karoubian envelopes, any functor CX −→ CY is represented as the composition of the canonical embedding CX −→ CXK and a uniquely Fe determined functor CXK −→ CY . If F is a (weakly) right ’exact’ functor from (CX , EX ) to (CY , EY ), then Fe is a (resp. weakly) right ’exact’ morphism from the Karoubian envelope

(CXK , EXK ) of (CX , EX ) to (CY , EY ).

1.5.5. Proposition. Let (CX , EX ) and (CY , EY ) be right exact categories. Suppose F that EX consists of split deflations. Then a functor CX −→ CY is a weakly right ’exact’ functor from (CX , EX ) to (CY , EY ) iff it maps deflations to deflations. e j Proof. Let M −→ N be a split epimorphism in CX and N −→ M its section. Set p = j ◦ e. Suppose that M ×N M exists (which is the case if e ∈ EX ). Then we have a commutative diagram p −−−→ e M −−−→ M −−−→ N id  M   t   id  id (1) y y M y N p1 −−−→ e M ×L M −−−→ M −−−→ N p2 F whose left vertical arrow, t, is uniquely determined. A functor CX −→ CY maps (1) to the commutative diagram F (p) −−−→ F (e) F (M) −−−→ F (M) −−−→ F (N) id       F (t) y y id y id (2) F (p1) −−−→ F (e) F (M ×L M) −−−→ F (M) −−−→ F (N) F (p2)

9 whose upper row is an exact diagram (by 1.5.1). Therefore, the lower row of (2) is an exact diagram. The assertion follows now from the deﬁnition of a weakly right ’exact’ functor.

1.5.6. Corollary. Let (CX , EX ) be a right exact category whose deﬂations are split. Then every presheaf of sets on (CX , EX ) is a sheaf. 2. Right exact categories with initial objects.

2.1. Kernels and cokernels of morphisms. Let CX be a category with an initial f object, x. For a morphism M −→ N we deﬁne the kernel of f as the upper horizontal arrow in a cartesian square k(f) Ker(f) −−−→ M   0  f y cart y f x −−−→ N when the latter exists. Cokernels of morphisms are deﬁned dually, via a cocartesian square

c(f) N −−−→ Cok(f) x x   0 f  cocart  f M −−−→ y where y is a ﬁnal object of CX . If CX is a pointed category (i.e. its initial objects are ﬁnal), then the notion of the f k(f) −−−→ kernel is equivalent to the usual one: the diagram Ker(f) −−−→ M −−−→ N is exact. 0 f −−−→ c(f) Dually, the cokernel of f makes the diagram M −−−→ N −−−→ Cok(f) exact. 0

2.1.1. Lemma. Let CX be a category with an initial object x. f (a) Let a morphism M −→ N of CX have a kernel. The canonical morphism k(f) Ker(f) −−−→ M is a monomorphism, if the unique arrow x −→iN N is a monomorphism. f (b) If M −→ N is a monomorphism, then x −→iM M is the kernel of f. Proof. (a) By deﬁnition of the kernel of f, we have a cartesian square

k(f) Ker(f) −−−→ M   0  f y cart y f iN x −−−→ N

k(f) Therefore, Ker(f) −−−→ M is a monomorphism if x −→iN N is a monomorphism.

10 f (b) Suppose that M −→ N is a monomorphism. If

φ L −−−→ x     ψ y y iN f M −−−→ N is a commutative square, then f equalizes the pair of arrows (ψ, iM ◦φ). If f is a monomorphism, the latter implies that ψ = iM ◦ φ. Therefore, in this case, the square

idx x −−−→ x     iM y y iN f M −−−→ N is cartesian.

2.1.2. Corollary. Let CX be a category with an initial object x. The following conditions are equivalent: f k(f) (a) If M −→ N has a kernel, then the canonical arrow Ker(f) −−−→ M is a monomorphism. iM (b) The unique arrow x −→ M is a monomorphism for any M ∈ ObCX . Proof. (a) ⇒ (b), because, by 2.1.1(b), the unique morphism x −→iM M is the kernel of the identical morphism M −→ M. The implication (b) ⇒ (a) follows from 2.1.1(a). 2.1.3. Note. The converse assertion is not true in general: a morphism might have a trivial kernel without being a monomorphism. It is easy to produce an example in the category of pointed sets. 2.2. Examples.

2.2.1. Kernels of morphisms of unital k-algebras. Let CX be the category Algk of associative unital k-algebras. The category CX has an initial object – the k-algebra k. ϕ For any k-algebra morphism A −→ B, we have a commutative square ϕ A −−−→ B x x   k(ϕ)   (ϕ) k ⊕ K(ϕ) −−−→ k where K(ϕ) denote the kernel of the morphism ϕ in the category of non-unital k-algebras and the morphism k(ϕ) is determined by the inclusion K(ϕ) −→ A and the k-algebra structure k −→ A. This square is cartesian. In fact, if ϕ A −−−→ B x x   γ   ψ C −−−→ k

11 is a commutative square of k-algebra morphisms, then C is an augmented algebra: C = k ⊕ K(ψ). Since the restriction of ϕ ◦ γ to K(ψ) is zero, it factors uniquely through K(ϕ). β Therefore, there is a unique k-algebra morphism C = k ⊕ K(ψ) −→ Ker(ϕ) = k ⊕ K(ϕ) such that γ = k(ϕ) ◦ β and ψ = (ϕ) ◦ β. ϕ This shows that each (unital) k-algebra morphism A −→ B has a canonical kernel Ker(ϕ) equal to the augmented k-algebra corresponding to the ideal K(ϕ).

k(ϕ) It follows from the description of the kernel Ker(ϕ) −−−→ A that it is a monomorphism iﬀ the k-algebra structure k −→ A is a monomorphism.

Notice that cokernels of morphisms are not deﬁned in Algk, because this category does not have ﬁnal objects.

2.2.2. Kernels and cokernels of maps of sets. Since the only initial object of the category Sets is the empty set ∅ and there are no morphisms from a non-empty set to ∅, the f kernel of any map X −→ Y is ∅ −→ X. The cokernel of a map X −→ Y is the projection c(f) Y −−−→ Y/f(X), where Y/f(X) is the set obtained from Y by the contraction of f(X) into a point. So that c(f) is an isomorphism iﬀ either X = ∅, or f(X) is a one-point set.

∧ 2.2.3. Presheaves of sets. Let CX be a svelte category and CX the category of non-trivial presheaves of sets on CX (that is we exclude the trivial presheaf which assigns ∧ to every object of CX the empty set). The category CX has a final object which is the constant presheaf with values in a one-element set. If CX has a final object, y, then ∧ ∧ yb = CX (−, y) is a final object of the category CX . Since CX has small colimits, it has cokernels of arbitrary morphisms which are computed object-wise, that is using 2.2.2. If the category CX has an initial object, x, then the presheaf xb = CX (−, x) is an ∧ ∧ initial object of the category CX . In this case, the category CX has kernels of all its ∧ h ∧ morphisms (because CX has limits) and the Yoneda functor CX −→ CX preserves kernels. ∧ Notice that the initial object of CX is not isomorphic to its final object unless the category CX is pointed, i.e. initial objects of CX are its final objects.

2.2.4. Sheaves of sets. Let τ be a pretopology on CX and CXτ denote the category ∧ of sheaves of sets on (CX , τ). Similarly to CX , the category CXτ has a ﬁnal object. If CX has an initial object x, then the sheaf associated with the presheaf CX (−, x) is an initial object of CXτ . In particular, CX (−, x) is an initial object of CXτ if it is a sheaf (say, the pretopology τ is subcanonical).

2.3. Some properties of kernels. Fix a category CX with an initial object x.

f 2.3.1. Proposition. Let M −→ N be a morphism of CX which has a kernel pair, p −→1 M ×N M −→ M. Then the morphism f has a kernel iﬀ p1 has a kernel. p2

12 Proof. Suppose that f has a kernel, i.e. there is a cartesian square

k(f) Ker(f) −−−→ M   0  f y y f (1) iN x −−−→ N

Then we have the commutative diagram

γ p2 Ker(f) −−−→ M ×N M −−−→ M    0    f y p1y y f (2) iM f x −−−→ M −−−→ N which is due to the commutativity of (1) and the fact that the unique morphism x −→iN N f factors through the morphism M −→ N. The morphism γ is uniquely determined by the equality p2 ◦ γ = k(f). The fact that the square (1) is cartesian and the equalities p2 ◦ γ = k(f) and iN = f ◦ iM imply that the left square of the diagram (2) is cartesian, γ i.e. Ker(f) −−−→ M ×N M is the kernel of the morphism p1. Conversely, if p1 has a kernel, then we have a diagram

k(p1) p2 Ker(p1) −−−→ M ×N M −−−→ M    0    p1 y cart p1y cart y f iM f x −−−→ M −−−→ N which consists of two cartesian squares. Therefore the square

k(f) Ker(p1) −−−→ M   0   p1y cart y f iN x −−−→ N with k(f) = p2 ◦ k(p1) is cartesian. 2.3.2. Remarks. (a) Needless to say that the picture obtained in (the argument of) 0 τf 2.3.1 is symmetric, i.e. there is an isomorphism Ker(p1) −→ Ker(p2) which is an arrow in the commutative diagram

k(p1) p1 Ker(p1) −−−→ M ×N M −−−→ M    τ 0 o τ o  id f y f y y M k(p2) p2 Ker(p2) −−−→ M ×N M −−−→ M

13 p f −→1 (b) Let a morphism M −→ N have a kernel pair, M ×N M −→ M, and a kernel. Then, p2 k(p1) by 2.3.1, there exists a kernel of p1, so that we have a morphism Ker(p1) −−−→ M ×N M ∆M and the diagonal morphism M −−−→ M ×N M. Since the left square of the commutative diagram 0 p1 x −−−→ Ker(p1) −−−→ x       y cart c(p1)y y ∆M p1 M −−−→ M ×N M −−−→ M is cartesian and compositions of the horizontal arrows are identical morphisms, it follows that its left square is cartesian too. Loosely, one can say that the intersection of Ker(p1) ` with the diagonal of M ×N M is zero. If there exists a coproduct Ker(p1) M, then the k(p1) ∆M pair of morphisms Ker(p1) −−−→ M ×N M ←−−− M determine a morphism a Ker(p1) M −−−→ M ×N M.

If the category CX is additive, then this morphism is an isomorphism, or, what is the ` same, Ker(f) M ' M ×N M. In general, it is rarely the case, as the reader can ﬁnd out looking at the examples of 2.2. 2.3.3. Proposition. Let f M −−−→e N f e   ge y cart y g (3) f M −−−→ N be a cartesian square. Then Ker(f) exists iﬀ Ker(fe) exists, and they are naturally isomorphic to each other. k(f) Proof. Suppose that Ker(f) −−−→ M exists, i.e. we have a cartesian square

k(f) Ker(f) −−−→ M   0  f y cart y f (4) iN x −−−→ N

g Since x −→ N factors through N 0 −→ N and the square (3) is cartesian, there is a unique γ morphism Ker(f 0) −→ N 0 such that the diagram

γ eg Ker(f) −−−→ Mf −−−→ M    0   (5) f y y fe cart y f g x −−−→ Ne −−−→ N

14 commutes and k(f) = ge ◦ γ. Therefore the left square of (5) is cartesian. If Ker(fe) exists, then we have the diagram

k(fe) eg Ker(fe) −−−→ Mf −−−→ M    0   fe y y fe cart y f g x −−−→ Ne −−−→ N whose both squares are cartesian. Therefore, their composition

eg◦k(fe) Ker(fe) −−−→ M   0  fe y y f iN x −−−→ N is a cartesian square. g0 It follows that the unique morphism Ker(fe) −→ Ker(f) making the diagram

k(fe) fe Ker(fe) −−−→ Mf −−−→ Ne    0   (6) g y ge y cart y g k(f) f Ker(f) −−−→ M −−−→ N commute is an isomorphism.

2.3.4. The kernel of a composition and related facts. Fix a category CX with an initial object x. f g 2.3.4.1. The kernel of a composition. Let L −→ M and M −→ N be morphisms such that there exist kernels of g and g ◦ f. Then the argument similar to that of 2.3.3 shows that we have a commutative diagram

f g0 Ker(gf) −−−→e Ker(g) −−−→ x       (1) k(gf) y cart y k(g) cart y iN f g L −−−→ M −−−→ N whose both squares are cartesian and all morphisms are uniquely determined by f, g and the (unique up to isomorphism) choice of the objects Ker(g) and Ker(gf). Conversely, if there is a commutative diagram

u g0 K −−−→ Ker(g) −−−→ x       t y cart y k(g) y iN f g L −−−→ M −−−→ N

15 whose left square is cartesian, then its left vertical arrow, K −→t L, is the kernel of the g◦f composition L −−−→ N. f 2.3.4.2. Remarks. (a) It follows from 2.3.3 that the kernel of L −→ M exists iﬀ f the kernel of Ker(gf) −−−→e Ker(g) exists and they are isomorphic to each other. More precisely, we have a commutative diagram

k(fe) fe g0 Ker(fe) −−−→ Ker(gf) −−−→ Ker(g) −−−→ x         oy k(gf) y cart y k(g) cart y iN k(f) f g Ker(f) −−−→ L −−−→ M −−−→ N whose left vertical arrow is an isomorphism. (b) Suppose that (CX , EX ) is a right exact category (with an initial object x). If the morphism f above is a deﬂation, then it follows from this observation that the canonical f morphism Ker(gf) −−−→e Ker(g) is a deﬂation too. In this case, Ker(f) exists, and we have a commutative diagram

k(fe) fe Ker(fe) −−−→ Ker(gf) −−−→ Ker(g)       oy k(gf) y cart y k(g) k(f) f Ker(f) −−−→ L −−−→ M whose rows are conﬂations. The following observations is useful (and will be used) for analysing diagrams. g 2.3.4.3. Proposition.(a) Let M −→ N be a morphism with a trivial kernel. Then f a morphism L −→ M has a kernel iﬀ the composition g ◦ f has a kernel, and these two kernels are naturally isomorphic one to another. (b) Let f L −−−→ M     γ y y g φ Mf −−−→ N be a commutative square such that the kernels of the arrows f and φ exist and the kernel of g is trivial. Then the kernel of the composition φ ◦ γ is isomorphic to the kernel of the morphism f, and the left square of the commutative diagram

∼ k(f) f Ker(f) −−−→ Ker(φγ) −−−→ L −−−→ M       γe y cart γ y y g k(φ) φ Ker(φ) −−−→ Mf −−−→ N

16 is cartesian.

Proof. (a) Since the kernel of g is trivial, the diagram 2.3.4.1(1) specializes to the diagram

f idx Ker(gf) −−−→e x −−−→ x       k(gf) y cart y k(g) y iN f g L −−−→ M −−−→ N with cartesian squares. The left cartesian square of this diagram is the deﬁnition of Ker(f). The assertion follows from 2.3.4.1. (b) Since the kernel of g is trivial, it follows from (a) that Ker(f) is naturally isomorphic to the kernel of g ◦ f = φ ◦ γ. The assertion follows now from 2.3.4.1.

f 2.3.4.4. Corollary. Let CX be a category with an initial object x. Let L −→ M be g k(g) a strict epimorphism and M −→ N a morphism such that Ker(g) −−−→ M exists and is a monomorphism. Then the composition g ◦ f is a trivial morphism iﬀ g is trivial.

Proof. The morphism g ◦ f being trivial means that there is a commutative square

f L −−−→ M     γ y y g iN x −−−→ N

By 2.3.4.3(a), Ker(g ◦ f) ' Ker(γ) = L. Thus, we have a commutative diagram

f g0 Ker(gf) −−−→e Ker(g) −−−→ x       oy cart y k(g) cart y iN f g L −−−→ M −−−→ N

(cf. 2.3.4.1). Since f is a strict epimorphism, it follows from the commutativity of the k(g) left square that Ker(g) −−−→ M is a strict epimorphism. Since, by hypothesis, k(g) is a monomorphism, it is an isomorphism, which implies the triviality of g.

2.3.4.4.1. Note. The following example shows that the requirement ”Ker(g) −→ M is a monomorphism” in 2.3.4.4 cannot be omitted. Let CX be the category Algk of associative unital k-algebras, and let m be an ideal of the ring k such that the epimorphism k −→ k/m does not split. Then the identical morphism k/m −→ k/m is non-trivial, while its composition with the projection k −→ k/m is a trivial morphism.

17 f 2.3.5. The coimage of a morphism. Let M −→ N be an arrow which has a kernel, i.e. we have a cartesian square

k(f) Ker(f) −−−→ M   0  f y cart y f iN x −−−→ N

k(f) −→ with which one can associate a pair of arrows Ker(f) −→ M, where 0f is the composition 0f 0 iM of the projection f and the unique morphism x −→ M. Since iN = f ◦ iM , the morphism k(f) −→ f equalizes the pair Ker(f) −→ M. If the cokernel of this pair of arrows exists, it will 0f be called the coimage of f and denoted by Coim(f), or. loosely, M/Ker(f). f Let M −→ N be a morphism such that Ker(f) and Coim(f) exist. Then f is the pf composition of the canonical strict epimorphism M −−−→ Coim(f) and a uniquely deﬁned jf morphism Coim(f) −−−→ N.

f 2.3.5.1. Lemma. Let M −→ N be a morphism such that Ker(f) and Coim(f) ∼ exist. There is a natural isomorphism Ker(f) −→ Ker(pf ). Proof. The outer square of the commutative diagram

f 0 Ker(f) −−−→ x −−−→ x       k(f)y cart y y (1) pf jf M −−−→ Coim(f) −−−→ L is cartesian. Therefore, its left square is cartesian which implies, by 2.3.3, that Ker(pf ) is isomorphic to Ker(f 0). But, Ker(f 0) ' Ker(f). 2.3.5.2. Note. By 2.3.4.1, all squares of the commutative diagram

f 0 Ker(f) −−−→ x     id y cart y epf (2) Ker(jf pf ) −−−→ Ker(jf ) −−−→ x       k(f)y cart y cart y pf jf M −−−→ Coim(f) −−−→ L are cartesian.

18 f If CX is an additive category and M −→ L is an arrow of CX having a kernel and jf a coimage, then the canonical morphism Coim(f) −−−→ L is a monomorphism. Quite a few non-additive categories have this property.

2.3.5.3. Example. Let CX be the category Algk of associative unital k-algebras. Since cokernels of pairs of arrows exist in Algk, any algebra morphism has a coimage. It ϕ follows from 2.2.1 that the coimage of an algebra morphism A −→ B is A/K(ϕ), where K(ϕ) is the kernel of φ in the usual sense (i.e. in the category of non-unital algebras). The canonical decomposition ϕ = jϕ ◦ pϕ coincides with the standard presentation of ϕ as the composition of the projection A −→ A/K(ϕ) and the monomorphism A/K(ϕ) −→ B. In particular, ϕ is strict epimorphism of k-algebras iff it is isomorphic to the associated pϕ coimage map A −−−→ Coim(ϕ) = A/K(ϕ). 2.4. Conflations and fully exact subcategories of a right exact category. Fix a right exact category (CX , EX ) with an initial object x. We denote by EX the class k e k of all sequences of the form K −→ M −→ N, where e ∈ EX and K −→ M is a kernel of e. Expanding the terminology of exact additive categories, we call such sequences conflations. 2.4.1. Fully exact subcategories of a right exact category. We call a full subcategory B of CX a fully exact subcategory of the right exact category (CX , EX ), if B contains the initial object x and is closed under extensions; i.e. if objects K and N in a conflation K −→k M −→e N belong to B, then M is an object of B. In particular, fully exact subcategories of (CX , EX ) are strictly full subcategories.

2.4.2. Proposition. Let (CX , EX ) be a right exact category with an initial object x e and B its fully exact subcategory. Then the class EX,B of all deﬂations M −→ N such that M,N, and Ker(e) are objects of B is a structure of a right exact category on B such that the inclusion functor B −→ CX is an ’exact’ functor (B, EX,B) −→ (CX , EX ).

Proof. (a) We start with the invariance of EX,B under base change. Let

e M −−−→e N f e   ge y cart y g e M −−−→ N be a cartesian square such that e (hence e) is a deﬂation and the objects M, N, Ker(e), and Ne belong to B. The claim is that the remaining object, Mf, belongs to B. In fact, consider the diagram

k(ee) ee Ker(e) −−−→ Mf −−−→ Ne e   0   (7) g y ge y cart y g k(e) e Ker(e) −−−→ M −−−→ N

19 Since its right square is cartesian, it follows from 2.3.3 that the canonical morphism g0 Ker(e) −−−→ Ker(e) is an isomorphism; i.e. the upper row of the diagram (7) is a conﬂation whose ends, Ker(e) and Ne, are objects of B. Since B is fully exact, the middle e object, Mf, belongs to B, which means that the deﬂation Mf −→e Ne belongs to EX,B. (b) The invariance of EX,B under base change implies that it is closed under com- s t position. In fact, let L −→ M −→ N be morphisms of EX,B. By 2.3.4.1, we have a commutative diagram

s t0 Ker(ts) −−−→e Ker(s) −−−→ x       (8) k(ts) y cart y k(s) cart y iN s t L −−−→ M −−−→ N

k(s) whose squares are cartesian. Since s belongs to EX,B, its kernel Ker(s) −−−→ M is an k(ts) arrow of B. Applying (a) to the left cartesian square of (8), we obtain that Ker(ts) −−−→ L is an arrow of B, which means that ts ∈ EX,B. (c) Each isomorphism of the category B belongs to the class EX,B, because each isomorphism is a deﬂation and its kernel is an initial object, and, by hypothesis, initial objects belong to B.

2.4.3. Remark. Let (CX , EX ) be a right exact category with an initial object x and B its strictly full subcategory containing x. Let E be a right exact structure on B such J that the inclusion functor B −→ CX maps deflations to deflations and preserves kernels of deflations. Then E is contained in EX,B. In particular, E is contained in EX,B if the inclusion functor is an ’exact’ functor from (B, E) to (CX , EX ). This shows that if B is a fully exact subcategory of (CX , EX ), then EX,B is the finest right exact structure on B such that the inclusion functor B −→ CX is an exact functor from (B, EX,B) to (CX , EX ).

2.5. Proposition. Let (CX , EX ) and (CY , EY ) be right exact categories and F a functor CX −→ CY which maps conﬂations to conﬂations. Suppose that the category CY is additive. Then the functor F is ’exact’.

Proof. Let F be a functor CX −→ CY which preserves conflations. We need to show that the functor F preserves arbitrary pull-backs of deflations. e f (a) Let M −→ N be a deflation and Ne −→ N a morphism of CX . Consider the associated with this data diagram

k(ee) ee Ker(e) −−−→ Mf −−−→ Ne e   00   (3) f y fe y cart y f k(e) e Ker(e) −−−→ M −−−→ N

20 whose right square is cartesian. Therefore, by 2.3.3, the left vertical arrow of the diagram f 00 (3), Ker(e) −→ Ker(e), is an isomorphism. Since the rows of the diagram (3) are conﬂa- tions and, by hypothesis, F preserves conﬂations, the rows of the commutative diagram

F (k(ee)) F (ee) F (Ker(e)) −−−→ F (Mf) −−−→ F (Ne)    00    F (f ) yo F (fe) y y F (f) (4) F (k(e)) F (e) F (Ker(e)) −−−→e F (M) −−−→ F (N) are conﬂations. The diagram (4) can be decomposed into a commutative diagram

F (k(ee)) F (ee) F (Ker(e)) −−−→ F (Mf) −−−→ F (Ne)       β y γ y y id k(e00) e00 00 (5) Ker(e ) −−−→ M −−−→ F (Ne)       ψ y φ y cart y F (f) F (k(e)) F (e) F (Ker(e)) −−−→e F (M) −−−→ F (N) where the right lower square is cartesian, γ is a morphism uniquely determined by the 00 00 equalities e ◦ γ = F (e) and φ ◦ γ = F (fe); and ψ ◦ β = F (f ). Since the lower row ψ of (5) is a conﬂation, it follows from 2.3.3 that the morphism Ker(e00) −→ F (Ker(e)) is an isomorphism. Therefore, β = ψ−1 ◦ F (f 00) is an isomorphism. Thus, we have a commutative diagram

F (k(ee)) F (ee) F (Ker(e)) −−−→ F (Mf) −−−→ F (Ne)       β yo γ y oy id (6) k(e00) e00 Ker(e00) −−−→ M −−−→ F (Ne) whose rows are conﬂations and two vertical arrows are isomorphisms. γ (b) The claim is that then the third vertical arrow, F (Mf) −→ M, is an isomorphism.

In fact, applying the canonical ’exact’ embedding of (CY , EY ) to the category CYE of sheaves of Z-modules on the presite (CY , EY ), we reduce the assertion to the case when the category is abelian (with the canonical exact structure); and the fact is well known for the abelian categories.

2.5.1. Corollary. Let (CX , EX ) and (CY , EY ) be additive k-linear right exact categories and F an additive functor CX −→ CY . Then the functor F is weakly ’exact’ iff it is ’exact’. F Proof. By 1.4, a k-linear functor CX −→ CY is a weakly ’exact’ iff it maps conflations to conflations. The assertion follows now from 2.5.

21 2.5.2. The property (†). In Proposition 2.5, the assumption that the category CY is additive is used only at the end of the proof (part (b)). Moreover, additivity appears there only because it garantees the following property: (†) if the rows of a commutative diagram

L −−−→ M −−−→ N e f e    y y y L −−−→ M −−−→ N are conﬂations and its right and left vertical arrows are isomorphisms, then the middle arrow is an isomorphism.

So that the additivity of CY in 2.5 can be replaced by the property (†) for (CY , EY ). 2.5.3. An observation. The following obvious observation helps to establish the property (†) for many non-additive right exact categories: F If (CX , EX ) and (CY , EY ) are right exact categories and CX −→ CY is a conservative functor which maps conﬂations to conﬂations, then the property (†) holds in (CX , EX ) provided it holds in (CY , EY ).

2.5.3.1. Example. Let (CY , EY ) are right exact k-linear category, (CX , EX ) a F right exact category, and CX −→ CY is a conservative functor which maps conflations to conflations. Then the property (†) holds in (CX , EX ). s For instance, the property (†) holds for the right exact category (Algk, E ) of associative unital k-algebras with strict epimorphisms as deflations, because the forgetful functor f∗ Algk −→ k − mod is conservative, maps deflations to deflations (that is to epimorphisms) and is left exact. Therefore, it maps conflations to conflations. 2.6. Digression: right exact additive categories and exact categories.

2.6.1. Proposition. Let (CX , EX ) be an additive k-linear right exact category.

Then there exists an exact category (CXe , EXe ) and a fully faithful k-linear ’exact’ functor ∗ γX (CX , EX ) −−−→ (CXe , EXe ) which is universal; that is any ’exact’ k-linear functor from ∗ (CX , EX ) to an exact k-linear category factorizes uniquely through γX .

Proof. We take as CXe the smallest fully exact subcategory of the category CXE of sheaves of k-modules on (CX , EX ) containing all representable sheaves. By 8.3.1, the sub- (∞) category CXe coincides with CeX , where CeX denotes the image of CX in CXE . Therefore, objects of the category CXe are sheaves F such that there exists a ﬁnite ﬁltration

0 = F0 −→ F1 −→ ... −→ Fn = F

such that Fm/Fm−1 is representable for 1 ≤ m ≤ n. By K5.1, the subcategory CXe , being a fully exact subcategory of an abelian category, is exact.

22 ϕ∗ Let (CY , EY ) be an exact k-linear category and (CX , EX ) −→ (CY , EY ) an exact k- linear functor. The functor ϕ∗ extends to a continuous (i.e. having a right adjoint) functor ∗ ϕe CXE −→ CYE such that the diagram

ϕ∗ C −−−→ C X Y ∗   ∗ jX y y jY ∗ ϕe CXE −−−→ CYE is quasi-commutative (see 1.2). Since the functor ϕ∗ is ’exact’, it preserves pullbacks of deﬂations. In particular, it preserves kernels of deﬂations. Therefore, the restriction of ∗ ϕ to the Gabriel square, CX(2) , of CX regarded as a subcategory of the exact category

(CXe , EXe ), preserves conﬂations, hence it is ’exact’. This implies that the restriction of ∗ ϕ to the n-th Gabriel power CX(n) , of CX (in (CXe , EXe )) is ’exact’ for all n, whence the assertion. 2.6.2. The bicategories of exact and right exact k-linear categories. Right exact svelte k-linear categories are objects of a bicategory Rexk. Its 1-morphisms are right weakly ’exact’ k-linear functors and 2-morphisms are morphisms between those functors. We denote by Exrk the full subbicategory of Rexk whose objects are exact k-linear categories. It follows from 2.6.1 that the inclusion functor Exrk −→ Rexk has a left adjoint (in the bicategorical sense). 3. Satellites in right exact categories. 3.1. Preliminaries: trivial morphisms, pointed objects, and complexes. Let CX be a category with initial objects. We call a morphism of CX trivial if it factors through an initial object. It follows that an object M is initial iﬀ idM is a trivial morphism. If CX is a pointed category, then the trivial morphisms are usually called zero morphisms. 3.1.1. Trivial compositions and pointed objects. If the composition of arrows f g L −→ M −→ N is trivial, i.e. there is a commutative square

f L −−−→ M     ξ y y g iN x −−−→ N where x is an initial object, and the morphism g has a kernel, then f is the composition of k(g) f the canonical arrow Ker(g) −→ M and a morphism L −→g Ker(g) uniquely determined by f and ξ. If the arrow x −→iN N is a monomorphism, then the morphism ξ is uniquely determined by f and g ; therefore in this case, the arrow fg does not depend on ξ.

3.1.1.1. Pointed objects. In particular, fg does not depend on ξ, if N is a pointed object. The latter means that therre exists an arrow N −→ x.

23 3.1.2. Complexes. A sequence of arrows

fn+1 fn fn−1 fn−2 ... −−−→ Mn+1 −−−→ Mn −−−→ Mn−1 −−−→ ... (1) is called a complex if each its arrow has a kernel and the next arrow factors uniquely through this kernel. 3.1.3. Lemma. Let each arrow in the sequence

f3 f2 f1 f0 ... −−−→ M3 −−−→ M2 −−−→ M1 −−−→ M0 (2) of arrows have a kernel and the composition of any two consecutive arrows is trivial. Then

f4 f3 f2 ... −−−→ M4 −−−→ M3 −−−→ M2 (3) is a complex. If M0 is a pointed object, then (2) is a complex.

f0◦f1 Proof. The composition M2 −−−→ M0 factors through an initial object; in particular, there exist morphisms from Mi to an initial object x of CX for all i ≥ 2. Therefore, the unique morphism x −→ Mi is a (split) monomorphism for all i ≥ 2. By 2.1.1(a), this k(fi) implies that Ker(fi) −−−→ Mi+1 is a monomorphism. Therefore, there exists a unique 0 fi+1 k(fi) arrow Mi+2 −−−→ Ker(fi) whose composition with Ker(fi) −−−→ Mi+1 equals to fi+1. By the similar reason, if there exists a morphism from M0 (resp. M1) to x, then k(fi) Ker(fi) −−−→ Mi+1 is a monomorphism for i ≥ 0 (resp. for i ≥ 1). 3.1.4. Corollary. A sequence of morphisms

fn+1 fn fn−1 fn−2 ... −−−→ Mn+1 −−−→ Mn −−−→ Mn−1 −−−→ ... unbounded on the right is a complex iﬀ the composition of any pair of its consecutive arrows is trivial and for every i, there exists a kernel of the morphism fi.

3.1.4.1. Example. Let CX be the category Algk of unital associative k-algebras. The algebra k is its initial object, and every morphism of k-algebras has a kernel. Pointed objects of CX which have a morphism to initial object are precisely augmented k-algebras. If the composition of pairs of consecutive arrows in the sequence

f3 f2 f1 f0 ... −−−→ A3 −−−→ A2 −−−→ A1 −−−→ A0 is trivial, then it follows from the argument of 3.1.2 that Ai is an augmented k-algebra for all i ≥ 2. And any unbounded on the right sequence of algebras with trivial compositions of pairs of consecutive arrows is formed by augmented algebras.

24 3.1.5. The categories of complexes. Let CX be a category with initial objects. For any integer m, we denote by Km(CX ) the category whose objects are complexes of the form fm+2 fm+1 fm ... −−−→ Mm+2 −−−→ Mm+1 −−−→ Mm and morphisms are deﬁned as usual. Every ﬁnite complex

fn−1 fn−2 fm+2 fm+1 fm Mn −−−→ Mn−1 −−−→ ... −−−→ Mm+2 −−−→ Mm+1 −−−→ Mm (3) is identiﬁed with an object of Km(CX ) by adjoining on the left the inﬁnite sequence of trivial objects and (unique) morphisms from them. We call an object (3) of the category Km(CX ) a bounded complex if Mn is an initial b object for all n m. We denote by Km(CX ) the full subcategory of Km(CX ) generated by bounded complexes. b The categories Km(CX ) (resp. Km(CX )) are naturally isomorphic to each other via obvious translation functors. We denote by K(CX ) the category whose objects are complexes

fn+1 fn fn−1 fn−2 ... −−−→ Mn+1 −−−→ Mn −−−→ Mn−1 −−−→ ... (4) which are inﬁnite in both directions. Unless CX is a pointed category, there are no natural embeddings of the categories Km(CX ) into K(CX ). There is a natural embedding into K(CX ) of the full subcategory Km,∗(CX ) of Km(CX ) generated by all complexes (3) with Mm equal to an initial object. We say that an object (4) of the category K(CX ) is a complex bounded on the left (resp. on the right) if Mn is an initial object for all n 0 (resp. n 0). We denote by + − K (CX ) (resp. by K (CX )) the full subcategory of K(CX ) whose objects are complexes b − T + bounded on the left (resp. on the right). Finally, we set K (CX ) = K (CX ) K (CX ) b and call objects of the subcategory K (CX ) bounded complexes.

3.1.6. ’Exact’ complexes. Let (CX , EX ) be a right exact category with an initial f g object. We call a sequence of two arrows L −→ M −→ N in CX ’exact’ if the arrow g k(g) f has a kernel, and f is the composition of Ker(g) −→ M and a deﬂation L −→g Ker(g). A complex is called ’exact’ if any pair of its consecutive arrows forms an ’exact’ sequence.

∗ 3.2. ∂ -functors. Fix a right exact category (CX , EX ) with an initial object x and ∗ a category CY with an initial object. A ∂ -functor from (CX , EX ) to CY is a system of Ti functors CX −→ CY , i ≥ 0, together with a functorial assignment to every conﬂation j e di(E) E = (N −→ M −→ L) and every i ≥ 0 a morphism Ti+1(L) −−−→ Ti(N) which depends functorially on the conﬂation E and such that the sequence of arrows

T2(e) d1(E) T1(j) T1(e) d0(E) T0(j) ... −−−→ T2(L) −−−→ T1(N) −−−→ T1(M) −−−→ T1(L) −−−→ T0(N) −−−→ T0(M)

25 id Ti(x) is a complex. Taking the trivial conflation x −→ x −→ x, we obtain that Ti(x) −−−→ Ti(x) is a trivial morphism, or, equivalently, Ti(x) is an initial object, for every i ≥ 1. 0 0 0 ∗ Let T = (Ti, di| i ≥ 0) and T = (Ti , di| i ≥ 0) be a pair of ∂ -functors from (CX , EX ) 0 fi 0 to CY . A morphism from T to T is a family f = (Ti −→ Ti | i ≥ 0) of functor morphisms j e such that for any conflation E = (N −→ M −→ L) of the exact category CX and every i ≥ 0, the diagram di(E) Ti+1(L) −−−→ Ti(N)     fi+1(L) y y fi(N) d0 (E) 0 i 0 Ti+1(L) −−−→ Ti (N) commutes. The composition of morphisms is naturally defined. Thus, we have the category ∗ ∗ Hom ((CX , EX ),CY ) of ∂ -functors from (CX , EX ) to CY . ∗ ∗ 3.2.1. Trivial ∂ -functors. We call a ∂ -functor T = (Ti, di| i ≥ 0) trivial if all ∗ Ti are functors with values in initial objects. One can see that trivial ∂ -functors are ∗ precisely initial objects of the category Hom ((CX , EX ),CY ). Once an initial object y of the category CY is fixed, we have a canonical trivial functor whose components equal to the constant functor with value in y – it maps all arrows of CX to idy.

3.2.2. Some natural functorialities. Let (CX , EX ) be a right exact category with an initial object and CY a category with initial object. If CZ is another category with an F initial object and CY −→ CZ a functor which maps initial objects to initial objects, then ∗ for any ∂ -functor T = (Ti, di| i ≥ 0), the composition F ◦ T = (F ◦ Ti,F di| i ≥ 0) of T with F is a ∂∗-functor. The map (F,T ) 7−→ F ◦ T is functorial in both variables; i.e. it extends to a functor

∗ ∗ Cat∗(CY ,CZ ) × Hom ((CX , EX ),CY ) −−−→ Hom ((CX , EX ),CZ ). (1)

Here Cat∗ denotes the subcategory of Cat whose objects are categories with initial objects and morphisms are functors which map initial objects to initial objects.

On the other hand, let (CX, EX) be another right exact category with an initial object and Φ a functor CX −→ CX which maps conflations to conflations. In particular, it maps idM initial objects to initial objects (because if x is an initial object of CX, then x −→ M −→ M id is a conflation; and Φ(x −→ M −→M M) being a conflation implies that Φ(x) is an initial ∗ object). For any ∂ -functor T = (Ti, di| i ≥ 0) from (CX , EX ) to CY , the composition ∗ T ◦Φ = (Ti ◦Φ, diΦ| i ≥ 0) is a ∂ -functor from (CX, EX) to CY . The map (T, Φ) 7−→ T ◦Φ extends to a functor

∗ ∗ Hom ((CX , EX ),CY ) × Ex∗((CX, EX), (CX , EX )) −−−→ Hom ((CX, EX),CY ), (2) where Ex∗((CX, EX), (CX , EX )) denotes the full subcategory of Hom(CX,CX ) whose objects are preserving conﬂations functors CX −→ CX .

26 ∗ 3.3. Universal ∂ -functors. Fix a right exact category (CX , EX ) with an initial object x and a category CY with an initial object y. ∗ ∗ A ∂ -functor T = (Ti, di| i ≥ 0) from (CX , EX ) to CY is called universal if for every ∂ - 0 0 0 0 g functor T = (Ti , di| i ≥ 0) from (CX , EX ) to CY and every functor morphism T0 −→ T0, 0 fi 0 there exists a unique morphism f = (Ti −→ Ti | i ≥ 0) from T to T such that f0 = g. 3.3.1. Interpretation. Consider the functor

Ψ∗ ∗ Hom ((CX , EX ),CY ) −−−→ Hom(CX ,CY ) (3) which assigns to every ∂∗-functor (resp. every morphism of ∂∗-functors) its zero compo- F ∗ nent. For any functor CX −→ CY , we have a presheaf of sets Hom(Ψ (−),F ) on the ∗ category Hom ((CX , EX ),CY ). Suppose that this presheaf is representable by an object ∗ ∗ (i.e. a ∂ -functor) Ψ∗(F ). Then Ψ∗(F ) is a universal ∂ -functor. ∗ Conversely, if T = (Ti, di| i ≥ 0) is a universal ∂ -functor, then T ' Ψ∗(T0).

3.3.2. Proposition. Let (CX , EX ) be a right exact category with an initial object x; and let CY be a category with initial objects, kernels of morphisms, and limits of ﬁltered F systems. Then, for any functor CX −→ CY , there exists a unique up to isomorphism ∗ universal ∂ -functor T = (Ti, di| i ≥ 0) such that T0 = F . In other words, the functor

Ψ∗ ∗ Hom ((CX , EX ),CY ) −−−→ Hom(CX ,CY ) (3)

∗ which assigns to each morphism of ∂ -functors its zero component has a right adjoint, Ψ∗. F Proof. (a) For an arbitrary functor CX −→ CY , we set S−(F )(L) = lim Ker(F (k(e))), where the limit is taken by the (filtered) system of all deflations M −→e L. Since deflations S−(F ) form a pretopology, the map L 7−→ S−(F )(L) extends naturally to a functor CX −−−→ CY . j e By the definition of S−(F ), for any conflation E = (N −→ M −→ L), there exists a unique ∂0 (E) eF 0 0 morphism S−(F )(L) −−−→ Ker(F (j)). We denote by ∂F (E) the composition of ∂eF (E) and the canonical morphism Ker(F (j)) −→ F (N). (b) Notice that the correspondence F 7−→ S−(F ) is functorial. Applying the iterations ∗ • i of the functor S− to F , we obtain a ∂ -functor S−(F ) = (S−(F )|i ≥ 0). The claim is that this ∂∗-functor is universal. ∗ λ0 In fact, let T = (Ti, d|i ≥ 0) be a ∂ -functor and T0 −→ F a functor morphism. For any conflation E = (N −→j M −→e L), we have a commutative diagram

d0(E) T0(j) T0(e) T1(L) −−−→ T0(N) −−−→ T0(M) −−−→ T0(L)       λ0(N) y y λ0(M) y λ0(L) (4) F (j) F (e) F (N) −−−→ F (M) −−−→ F (L)

27 d0(E) T0(j) Since T1(L) −−−→ T0(N) −−−→ T0(M) is a complex, the morphism d0(E) is the ed0(E) composition of a uniquely deﬁned morphism T1(L) −−−→ Ker(T0(j)) and the canonical arrow Ker(T0(j)) −→ T0(N). We denote by λe1(E) the composition of the morphism ed0(E) 0 λ1 and the morphism Ker(T0(j)) −−−→ Ker(F (j) uniquely determined by the commutativity of the diagram

k(T0(j)) T0(j) Ker(T0(j)) −−−→ T0(N) −−−→ T0(M)    0    λ1 y λ0(N) y y λ0(M) k(F (j)) F (j) Ker(F (j)) −−−→ F (N) −−−→ F (M) Thus, we have a commutative diagram

d0(E) T0(j) T0(e) T1(L) −−−→ T0(N) −−−→ T0(M) −−−→ T0(L)         λe1(E) y λ0(N) y y λ0(M) y λ0(L) k(F (j)) F (j) F (e) Ker(F (j)) −−−→ F (N) −−−→ F (M) −−−→ F (L) with the morphism λe1(E) uniquely determined by the arrows of the diagram (4). Since the d0(E) connecting morphism T1(L) −−−→ T0(N) depends on the conflation E functorially, same eλ1(E) is true for λe1(E); that is the morphisms T1(L) −−−→ Ker(F (j)), where E runs through conflations N −→ M −→ L (with fixed L and morphisms of the form (h, g, idL)), form λ1(L) a cone. This cone defines a unique morphism T1(L) −−−→ S−(F )(L). It follows from the universality of this construction that λ = (λ1(L)| L ∈ ObCX ) is a functor morphism λ1 T1 −−−→ S−(F ) such that the diagram

d0(E) T0(j) T0(e) T1(L) −−−→ T0(N) −−−→ T0(M) −−−→ T0(L)         λ1(L) y λ0(N) y y λ0(M) y λ0(L) k(F (j)) F (j) F (e) S−(F )(L) −−−→ F (N) −−−→ F (M) −−−→ F (L) commutes. Iterating this construction, we obtain uniquely deﬁned functor morphisms λ i i Ti −−−→ S−(F ) for all i ≥ 1. 3.3.3. Remark. Let the assumptions of 3.3.2 hold. Then we have a pair of adjoint functors

Ψ∗ Ψ ∗ ∗ ∗ Hom ((CX , EX ),CY ) −−−→ Hom(CX ,CY ) −−−→ Hom ((CX , EX ),CY )

∗ By 3.3.2, the adjunction morphism Ψ Ψ∗ −→ Id is an isomorphism which means that Ψ∗ is a fully faithful functor and Ψ∗ is a localization functor at a left multiplicative system.

28 3.3.4. Proposition. Let (CX , EX ) be a right exact category with an initial object and ∗ T = (Ti, di | i ≥ 0) a ∂ -functor from (CX , EX ) to CY . Let CZ be another category with an initial object and F a functor from CY to CZ which preserves initial objects, kernels of morphisms and limits of filtered systems. Then ∗ (a) If T is a universal ∂ -functor, then F ◦ T = (F ◦ Ti,F di| i ≥ 0) is universal. (b) If, in addition, the functor F is fully faithful, then the ∂∗-functor F ◦T is universal iff T is universal. ∗ Proof. (a) Suppose that the ∂ -functor T = (Ti, di | i ≥ 0) is universal. Then it i follows from the argument of 3.3.2 that Ti ' S−(T0) for all i ≥ 0, where S−(G)(L) = lim Ker(G(k(e))) and the limit is taken by the system of all deflations M −→e L. Since the functor F preserves kernels of morphisms and filtered limits (that is all types of limits which appear in the construction of S−(G)(L)), the natural morphism

F ◦ S−(G)(L) −→ S−(F ◦ G)(L)

G is an isomorphism for any functor CX −→ CY such that S−(G)(L) = lim Ker(G(k(e))) ex- i i ists. Therefore, the natural morphism F ◦S−(T0)(L) −→ S−(F ◦T0)(L) is an isomorphism for all i ≥ 0 and all L ∈ ObCX . (b) Suppose that the functor F is fully faithful and the ∂∗-functor F ◦ T is universal. Then F ◦ Ti+1(L) ' S−(F ◦ Ti)(L) = lim Ker(F ◦ Ti(k(e))) '

lim F (Ker(Ti(k(e)))) ' F (lim Ker(Ti(k(e)))) = F (S−(Ti)(L)), where the isomorphisms are due to compatibility of F with kernels of morphisms and ﬁltered limits. Since all these isomorphisms are natural (i.e. functorial in L), we obtain ∼ a functor isomorphism F ◦ Ti+1 −→ F ◦ S−(Ti). Since the functor F is fully faithful, the ∼ latter implies an isomorphism Ti+1 −→ S−(Ti) for all i ≥ 0. The assertion follows now from (the argument of) 3.3.2.

3.3.5. An application. Let (CX , EX ) be a right exact category and CY a category with an initial object. Consider the Yoneda embedding

h Y ∧ CY −−−→ CY ,M 7−→ Mc = CY (−,M).

∧ of the category CY into the category CY of presheaves of sets on CY . The functor hY is fully faithful and preserves all limits. In particular, it satisfies the conditions of 3.3.4(b). ∗ Therefore, a ∂ -functor T = (Ti, di | i ≥ 0) from (CX , EX ) to CY is universal iff the ∗ def ∧ ∂ -functor Tb = hY ◦ T = (Tbi, bdi | i ≥ 0) from (CX , EX ) to CY is universal. G ∧ By 3.3.2, for any functor CX −→ CY , there exists a unique up to isomorphism uni- ∗ versal ∂ -functor T = (Ti, di| i ≥ 0) = Ψ∗(G) such that T0 = G. In particular, for F ∗ every functor CX −→ CY , there exists a unique up to isomorphism universal ∂ -functor T = (Ti, di | i ≥ 0) such that T0 = hY ◦ F = Fe. It follows from 3.3.4(b) that there exists a ∗ universal ∂ -functor whose zero component coincides with F iff for all L ∈ ObCX and all i ≥ 1, the presheaves Ti(L) are representable.

29 3.3.6. Remark. Let (CX , EX ) be a svelte right exact category with an initial object x and CY a category with an initial object y and limits. Then, by the argument of 3.3.2, we have an endofunctor S− of the category Hom(CX ,CY ) of functors from CX to CY , λ together with a cone S− −→ y, where y is the constant functor with the values in the j e initial object y of the category CY . For any conﬂation E = (N −→ M −→ L) of (CX , EX ) F and any functor CX −→ CY , we have a commutative diagram λ(L) S−F (L) −−−→ y     d0(E) y y F j F e F (N) −−−→ F (M) −−−→ F (L)

3.3.7. Digression: deﬂations with trivial kernels. Let (CX , EX ) be a right exact ~ category with an initial objects. We denote by EX the class of all arrows of EX whose kernel is an initial object.

~ 3.3.7.1. Proposition. The class of arrows EX is a right exact structure on the category CX . ~ Proof. The class EX contains all isomorphisms of the category CX . It is closed under compositions, because, by 2.3.4.3, if Ker(s) is trivial (i.e is an initial object of CX ), then Ker(s ◦ t) is naturally isomorphic to Ker(t). In particular, Ker(s ◦ t) is trivial, if both Ker(s) and Ker(t) are trivial. Finally, if

p1 M −−−→ M f    t y y s f N −−−→ L

~ is a cartesian square, then, by 2.3.3, Ker(s) ' Ker(t), which shows that EX is stable under base change.

3.3.7.2. Proposition. Let (CX , EX ) be a right exact category with an initial object x; and let CY be a category with initial objects, kernels of morphisms, and limits of ﬁltered ∗ systems. Let T = (Ti, di | i ≥ 0) be a universal ∂ -functor from (CX , EX ) to CY . If the ~ functor T0 maps all arrows of EX to isomorphisms, then all functors Ti, i ≥ 0, have this property. Proof. By the argument of 3.3.2, the assertion is equivalent to the following one: F ~ If a functor CX −→ CY maps arrows of EX to isomorphisms, then its satellite, S−F , has the same property. s e ~ In fact, let L −→ L be an arrow of EX and M −→ L an arbitrary deﬂation. Then we have a commutative diagram k(ee) ee Ker(e) −−−→ Mf −−−→ L  e      (1) s2 yo s1 y cart y s k(e) e Ker(e) −−−→ M −−−→ L

30 ~ whose vertical arrows belong to EX . Therefore, the left square of (1) determines iso- φ(e) morphism Ker(F (k(e)) −−−→ Ker(F (k(e)) which is functorial in e. So that we obtain e ∼ an isomorphism lim Ker(F (k(e)) −−−→ lim Ker(F (k(e)) = S−F (L) whose composition with the canonical arrow S−F (L) −−−→ lim Ker(F (k(e)) coincides with the morphism S−F (s) S−F (L) −−−→ S−F (L) (see the argument of 3.3.2). γ On the other hand, for any deﬂation M0 −→ L, there is a commutative diagram k(γ) γ Ker(γ) −−−→ M1 −−−→ L       yo id y y s (2) k(sγ) sγ Ker(sγ) −−−→ M1 −−−→ L Here the left vertical arrow is an isomorphism, because Ker(s) is an initial object (see φ(γ) 2.3.4.3). The left square of (2) induces an isomorphism Ker(F (k(sγ)) −−−→ Ker(F (k(γ)) which is functorial in γ. The latter implies that the composition ϕ(γ) of φ(γ) with the ϕ(γ) unique morphism S−F (L) −−−→ Ker(F (k(sγ)) deﬁnes a cone S−F (L) −−−→ Ker(F (k(γ)), ϕ hence a unique morphism S−F (L) −−−→ S−F (L). The claim is that ϕ is the inverse to S−F (s) the morphism S−F (L) −−−→ S−F (L). We complete (2) to a commutative diagram k(seγ) seγ Ker(sγ) −−−→ Mf1 −−−→ L f      t2 yo t1 y y id k(γ) γ (3) Ker(γ) −−−→ M1 −−−→ L       s2 yo id y y s k(sγ) sγ Ker(sγ) −−−→ M1 −−−→ L where the square seγ Mf1 −−−→ L     t1 y y s sγ M1 −−−→ L ~ is cartesian. Since t1 ∈ EX , the diagram (3) induces isomorphisms ∼ ∼ KerF (k(sfγ)) −→ KerF (k(γ)) −→ KerF (k(sγ)) which imply isomorphisms of the lower row of the commutative diagram

id S−F (s) id S−F (L) −−−→ S−F (L) −−−→ S−F (L) ←−−− S−F (L)         y y id y ϕ y ∼ id ∼ lim KerF (k(sfγ)) −−−→ lim KerF (k(γ)) −−−→ S−F (L) −−−→ lim KerF (k(sγ)) 31 The isomorphness of ϕ (or, equivalently, the isomorphness of S−F (s)) follows from the universal property of limits.

3.4. The dual picture: ∂-functors and universal ∂-functors. Let (CX , IX ) be op op a left exact category, which means by deﬁnition that (CX , IX ) is a right exact category. ∗ A ∂-functor on (CX , IX ) is the data which becomes a ∂ -functor in the dual right exact ∗ category. A ∂-functor on (CX , IX ) is universal if its dualization is a universal ∂ -functor. We leave to the reader the reformulation in the context of ∂-functors of all notions and facts about ∂∗-functors. Below, there are two versions – non-linear and linear, of a fundamental example of a universal ∂-functor. • 3.4.1. Example: Ext . Let (CX , EX ) be a right exact category with an initial object. For any L ∈ ObCX , we have the corresponding representable functor

h (L) op X op CX −−−→ Sets, M 7−→ CX (L, M) = CX (M,L). • Therefore, by (the dual version of) 3.3.2, there exists a universal ∂-functor ExtX (L) = i 0 (ExtX (L)| i ≥ 0), whose zero component, ExtX (L), coincides with hX (L). • 3.4.2. The functors Extk. Suppose that the category CX is k-linear. Then for any L ∈ ObCX , the functor hX (L) factors through the category k − mod (that is through the forgetful functor k − mod −→ Sets). Therefore, by 3.3.2, there exists a universal ∂-functor • i 0 ExtX (L) = (ExtX (L)| i ≥ 0), whose zero component, ExtX (L), coincides with the presheaf of k-modules CX (−,L). 3.5. Universal ∂∗-functors and ’exactness’. ∗ 3.5.1. The properties (CE5) and (CE5 ). Let (CX , EX ) be a right exact category. We say that it satisfies (CE5∗) (resp. (CE5)) if the limit of a filtered system (resp. the colimit of a cofiltered system) of conflations in (CY , EY ) exists and is a conflation. ∗ In particular, if (CX , EX ) satisfies (CE5 ) (resp. (CE5)), then the limit of any filtered system (resp. the colimit of any cofiltered system) of deflations is a deflation. The properties (CE5) and (CE5∗) make sense for left exact categories as well. Notice that a right exact category satisfies (CE5∗) (resp. (CE5)) iff the dual left exact category satisfies (CE5) (resp. (CE5∗)).

3.5.2. Note. If (CX , EX ) is an abelian category with the canonical exact structure, then the property (CE5) for (CX , EX ) is equivalent to the Grothendieck’s property (AB5) and, therefore, the property (CE5∗) is equivalent to (AB5∗) (see [Gr, 1.5]). The property (CE5) holds for Grothendieck toposes. In what follows, we use (CE5∗) for right exact categories and the dual property (CE5) for left exact categories.

3.5.3. Proposition. Let (CX , EX ), (CY , EY ) be right exact categories, and (CY , EY ) ∗ satisfy (CE5 ). Let F be a weakly right ’exact’ functor (CX , EX ) −→ (CY , EY ) such that j e S−(F ) exists. Then for any conﬂation E = (N −→ M −→ L) in (CX , EX ), the sequence

S−(F )(j) S−(F )(e) d0(E) F (j) S−(F )(N) −−−→ S−(F )(M) −−−→ S−(F )(L) −−−→ F (N) −−−→ F (M) (1)

32 is ’exact’. The functor S−(F ) is a weakly right ’exact’ functor from (CX , EX ) to (CY , EY ). F Proof. Let CX −→ CY be a right ’exact’ functor such that S−(F ) exists. (a) The claim is that for any conflation E = (N −→j M −→e L), the canonical ed0(E) morphism S−(F )(L) −−−→ Ker(F (j)) is a deflation. 0 e 0 e (a1) Let M −→ L and M −→ L be deflations of an object L of CX , and let

f M 0 −−−→ M e0 &. e L be a commutative diagram (– a morphism of deﬂations). This diagram extends to a morphism of the corresponding conﬂations

j0 e0 N 0 −−−→ M 0 −−−→ L    0   f y cart y f y idL (2) j e N −−−→ M −−−→ L

Since e0 = e ◦ f, it follows from 2.3.4.1 that the left square of (2) is cartesian. F For an arbitrary functor CX −→ CY , the diagram (2) gives rise to the commutative diagram

k0 F (j0) F (e0) Ker(F (j0)) −−−→ F (N 0) −−−→ F (M 0) −−−→ F (L)         γe y cart γ y y id y id k(α) α F (e0) Ker(α) −−−→ N −−−→ F (M 0) −−−→ F (L) (3)         φe yo φ y cart y F (f) y id k F (j) F (e) Ker(F (j)) −−−→ F (N) −−−→ F (M) −−−→ F (L) where the lower middle square is cartesian which implies (by 2.3.3) that φe is an isomorphism; the morphism γ is uniquely determined by the equalities φ ◦ γ = F (f 0) and F (j0) = α ◦ γ, and the left upper square is cartesian due to the latter equality (see 2.3.4.1). f (a2) Suppose now that the morphism M 0 −→ M in the diagram (2) (and (3)) is a deflation and the functor F is right ’exact’. Since the left square of the diagram (2) is cartesian, the morphism γ in (3) is a deflation. Therefore, since the left upper square in (3) is cartesian, the arrow γe is a deflation; or, what is the same, the canonical morphism 0 Ker(F (j )) −→ Ker(F (j)) (equal to the composition φe ◦ γe) is a deflation. 0 0 (a3) Notice that S−(F )(L) is isomorphic to the limit of Ker(F (k(e ))), where e runs e through the (filtered) diagram EX /M of refinements of the deflation M −→ L. That is S−(F ) = lim Ker(F (k(t ◦ e))), where t runs through the deflations of M (and morphisms of this diagram are also deflations).

33 ed0(E) Thus, the canonical morphism S−(F )(L) −−−→ Ker(F (j)) is the limit of a filtered system of deflations. Therefore, by hypothesis, it is a deflation.

j e (b) For any conﬂation E = (N −→ M −→ L) of the right exact category (CX , EX ), the canonical morphism S−(F )(M) −→ Ker(d0(E)) is a deﬂation. In fact, let

j00 e00 N 00 −−−→ M 00 −−−→ M       k y y id y e j0 e0 N 0 −−−→e M 00 −−−→e L e   0   0  et y cart y t y id (4) j0 e0 N 0 −−−→ M 0 −−−→ L       et y cart y t y id j e N −−−→ M −−−→ L

be a commutative diagram whose rows are conflations and the morphisms t and t0 are deflations. By 2.3.4.1 (or 2.3.4.3) that the two lower left squares of (4) are cartesian. In particular, the arrows et0 and et are deflations. It follows from 2.3.4.2(b) that the upper two k arrows of the left column of (4) form a conflation; i.e. N 00 −→ N 0 is the kernel of et0. The diagram (4) yeilds the commutative diagram

F (j00) F (e00) Ker(F (j00)) −−−→ F (N 00) −−−→ F (M 00) −−−→ F (M)         y cart F (k) y y id y F (e) 0 0 F (ej ) F (ee ) Ker(F (ej0)) −−−→ F (Ne 0) −−−→ F (M 00) −−−→ F (L)      0   0  (5) y F (et ) y y F (t ) y id F (j0) F (e0) Ker(F (j0)) −−−→ F (N 0) −−−→ F (M 0) −−−→ F (L)         y F (et) y y F (t) y id F (j) F (e) Ker(F (j)) −−−→ F (N) −−−→ F (M) −−−→ F (L)

Since the functor F is weakly right ’exact’, the diagram (5) is decomposed into the

34 diagram

Ker(F (j00)) −−−→ F (N 00)     γ1y cart γ2y F (j00) F (e00) Ker(s) −−−→ Ker(F (et0)) −−−→ F (M 00) −−−→ F (M)      0    k(s) y cart k y y id y F (e) 0 0 F (ej ) F (ee ) Ker(F (ej0)) −−−→ F (Ne 0) −−−→ F (M 00) −−−→ F (L) (6)      0   0  s y F (et ) y y F (t ) y id F (j0) F (e0) Ker(F (j0)) −−−→ F (N 0) −−−→ F (M 0) −−−→ F (L)         y F (et) y y F (t) y id F (j) F (e) Ker(F (j)) −−−→ F (N) −−−→ F (M) −−−→ F (L)

0 0 0 00 where γ1, γ2 are deﬂations, k is the kernel (morphism) of F (et ); F (j ) ◦ γ2 = F (j ), and 0 k ◦ γ2 = F (k). It follows that the two upper left squares of (6) are cartesian. The left column of the diagram (6) induces, via passing to limit, the sequence of arrows

eγ k0 σ k(F (j)) S−F (M) −−−→ Ker(d0(E)) −−−→ S−F (L) −−−→ Ker(F (j)) −−−→ F (N) where k(F (j)) ◦ σ = d0(E), k0 ◦ γe = S−F (e); σ is a deflation by (a) above, and γe is a deflation by hypothesis, because it is a filtered limit of deflations. 3.6. ’Exact’ ∂∗-functors and universal ∂∗-functors. Fix right exact categories ∗ (CX , EX ) and (CY , EY ), both with initial objects. A ∂ -functor T = (Ti, di| i ≥ 0) from j e (CX , EX ) to CY is called ’exact’ if for every conflation E = (N −→ M −→ L) in (CX , EX ), the complex

T2(e) d1(E) T1(j) T1(e) d0(E) T0(j) ... −−−→ T2(L) −−−→ T1(N) −−−→ Ti(M) −−−→ T1(L) −−−→ T0(N) −−−→ T0(M) is ’exact’.

3.6.1. Proposition. Let (CX , EX ), (CY , EY ) be right exact categories. Suppose that ∗ ∗ (CY , EY ) satisﬁes (CE5 ). Let T = (Ti| i ≥ 0) be a universal ∂ -functor from (CX , EX ) ∗ to (CY , EY ). If the functor T0 is right ’exact’, then the universal ∂ -functor T is ’exact’.

Proof. If T0 is right ’exact’, then, by 3.5.3, the functor T1 ' S−(T0) is right ’exact’ and for any conﬂation E = (N −→j M −→e L), the sequence

T1(j) T1(e) d0(E) T0(j) T1(N) −−−→ T1(M) −−−→ T1(L) −−−→ T0(N) −−−→ T0(M) is ’exact’. Since Tn+1 = S−(Tn), the assertion follows from 3.5.3 by induction.

35 3.6.2. Corollary. Let (CX , EX ) be a right exact category. For each object L of CX , • i the ∂-functor ExtX (L) = (ExtX (L) | i ≥ 0) is ’exact’. Suppose that the category CX is k-linear. Then for each L ∈ ObCX , the ∂-functor • i ExtX (L) = (ExtX (L) | i ≥ 0) is ’exact’. • Proof. In fact, the ∂-functor ExtX (L) is universal by definition (see 3.4.1), and the 0 functor ExtX (L) = CX (−,L) is left exact. In particular, it is left ’exact’. • If CX is a k-linear category, then the universal ∂-functors ExtX (L),L ∈ ObCX , with the values in the category of k-modules (defined in 3.4.2) are ’exact’ by a similar reason. 3.7. Universal problems for universal ∂∗- and ∂-functors. Fix a right exact ∗ svelte category (CX , EX ) with an initial object. Let ∂ Un(X, EX ) denote the category ∗ whose objects are universal ∂ -functors from (CX , EX ) to categories CY (with initial ob- ∗ ∗ jects). Let T be a universal ∂ - functor from (CX , EX ) to CY and Te a universal ∂ - functor 0 from (CX , EX ) to CZ . A morphism from T to T is a pair (F, φ), where F is a functor from ∗ ∼ 0 CY to CZ which preserves filtered limits and φ is a ∂ -functor isomorphism F ◦ T −→ T . It (F 0, φ0) is a morphism from T 0 to T 00, then the composition of (F, φ) and (F 0, φ0) is defined by (F 0, φ0) ◦ (F, φ) = (F 0 ◦ F, φ0 ◦ F 0φ). ∗ ∗ ∗ We denote by ∂ Unc(X, EX ) the subcategory of ∂ Un(X, EX ) whose objects are ∂ - functors from (CX , EX ) to complete (i.e. having limits of small diagrams) categories CY and morphisms are pairs (F, φ) such that the functor F preserves limits.

Dually, for a left exact category (CX, IX) with a final object, we denote by ∂Un(X, IX) the category whose objects are universal ∂-functors from (CX, IX) to categories with final 0 object. Given two universal ∂-functors T and T from (CX, IX) to respectively CY and CZ , 0 a morphism from T to T is a pair (F, ψ), where F is a functor from CY to CZ preserving filtered colimits and ψ is a functor isomorphism T 0 −→∼ F ◦ T . The composition is defined by (F 0, ψ0) ◦ (F, ψ) = (F 0 ◦ F,F 0ψ ◦ ψ0). c We denote by ∂Un (X, IX) the subcategory of ∂Un(X, IX) whose objects are ∂-functors with values in cocomplete categories and morphisms are pairs (F, ψ) such that the functor F preserves colimits.

3.7.1. Proposition. Let (CX , EX ) be a svelte right exact category with initial objects ∗ and (CX, IX) a svelte left exact category with ﬁnal objects. The categories ∂ Un(X, EX ), ∗ c ∂ Unc(X, EX ), ∂Un(X, IX), and ∂Un (X, IX) have initial objects. c Proof. (a) We start with the category ∂Un (X, IX). Consider the Yoneda embedding

h X ∧ CX −−−→ CX ,M 7−→ CX(−,M).

We denote by Ext• the universal ∂-functor from (C , I ) to C∧ such that Ext0 = X,IX X X X X,IX h . The claim is that Ext• is an initial object of the category ∂Unc(X, I ). X X,IX X In fact, let CY be a cocomplete category. By [GZ, II.1.3], the composition with the hX ∧ ∧ Yoneda embedding CX −→ CX is an equivalence between the category Homc(CX,CY ) of continuous (that is having a right adjoint, or, equivalently, preserving colimits) functors ∧ F CX −→ CY and the category Hom(CX,CY ) of functors from CX to CY . Let CX −→ CY be

36 ∧ Fc an arbitrary functor and CX −→ CY the corresponding continuous functor. By definition, j S+F (L) = colim(Cok(F (M −→ Cok(j)), where L −→ M runs through inflations of L. Since F preserves colimits, it follows from (the dual version of) 3.3.4(a) that F ◦ Ext• c c X,IX is a universal ∂-functor whose zero component is F ◦ Ext0 = F ◦ h = F. Therefore, c X,IX c X by (the dual version of the argument of) 3.3.2, the universal ∂-functor F ◦ Ext• is iso- c X,IX morphic to S• F . This shows that Ext• is an initial object of the category ∂Unc(X, I ). + X,IX X ∧ (b) Let CXs denote the smallest strictly full subcategory of the category CX containing all presheaves Extn (L),L ∈ ObC , n ≥ 0. The category ∂Un(X, I ) has an initial X,IX X X object which is the corestriction of the ∂-functor Ext• to the subcategory C . X,IX Xs Indeed, let CY be a category with a final object and T = (Ti, di | i ≥ 0) a universal ∂- ∨op op functor from (CX, IX) to CY . Let CY denote the category of presheaves of sets on CY (i.e. o ∨op functors CY −→ Sets) and hY the (dual) Yoneda functor CY −→ CY ,L 7−→ CY (L, −). ∨op o The category CY is cocomplete (and complete) and the functor hY preserves colimits. o Therefore, by 3.3.4, the composition hY ◦ T is a universal ∂-functor from (CX, IX) to ∨op o CY . By (a) above, the ∂-functor hY ◦ T is the composition of the universal ∂-functor op Ext• from (C , I ) to C∧ and the unique continuous functor C∧ −→G C∨ such that X,IX X X X X Y o o G ◦ hX = hY ◦ T0. Since the functor hY is fully faithful, this implies that the universal ∂- functor T = (T , d | i ≥ 0) is isomorphic to the composition of the corestriction of Ext• i i X,IX Gs to the subcategory CXs and a unique functor CXs −−−→ CY such that the composition o hY ◦ Gs coincides with the restriction of the functor G to the subcategory CXs . (c) The assertions about ∂∗-functors are obtained via dualization. Essential details are ∨ as follows. Let (CX , EX ) be a right exact category with initial objects. We denote by CX the category of functors CX −→ Sets (interpreted as the category of presheaves of sets on op Cop) and by ho the (dual) Yoneda functor C −→ C∨ ,M 7−→ C(M, −). Let Ext• X X X X (X,EX ) op denote the universal ∂∗-functor from (C , E ) to C∨ such that Ext0 = ho . X X X (X,EX ) X Let CY be a complete category. By the dual version of [GZ, II.1.3], the composition o with the functor hX is a category equivalence between the category Hom(CX ,CY ) and the c ∨op ∨op category Hom (CX ,CY ) of functors CX −→ CY which have a left adjoint. Let F be c a functor CX −→ CY and F the corresponding cocontinuous (i.e. having a left adjoint) ∨op c functor from CX to CY . Since the functor F preserves limits, it follows from 3.3.4 (a), that the composition F c ◦ Ext• is a universal ∂∗-functor. Its zero component, (X,EX ) F c ◦ Ext0 = F c ◦ ho , equals to F . Therefore, by 3.3.2, the universal ∂∗-functor (X,EX ) X F c ◦ Ext• is isomorphic to S• F . This shows that Ext• is an initial object of (X,EX ) − (X,EX ) ∗ the category ∂ Unc(X, EX ). It follows from (b) (by duality) that the corestriction of the ∂∗-functor Ext• (X,EX ) ∨op to the smallest subcategory of CX containing all representable functors and closed un- ∨op der the endofunctor S− (that is the full subcategory of CX generated by the functors Extn (L),L ∈ ObC , n ≥ 0) is an initial object of the category ∂∗Un(X, E ) of (X,EX ) X X universal ∂∗-functors.

3.7.2. The k-linear version. Fix a right exact k-linear additive category (CX , EX ). ∗ ∗ Let ∂kUn(X, EX ) denote the category whose objects are universal k-linear ∂ -functors from ∗ (CX , EX ) to k-linear additive categories CY . Let T be a universal k-linear ∂ - functor

37 ∗ from (CX , EX ) to CY and Te a universal k-linear ∂ - functor from (CX , EX ) to CZ .A 0 morphism from T to T is a pair (F, φ), where F is a k-linear functor from CY to CZ which preserves filtered limits and φ is a ∂∗-functor isomorphism F ◦ T −→∼ T 0. It (F 0, φ0) is a morphism from T 0 to T 00, then the composition of (F, φ) and (F 0, φ0) is defined by (F 0, φ0) ◦ (F, φ) = (F 0 ◦ F, φ0 ◦ F 0φ). ∗ ∗ We denote by ∂kUnc(X, EX ) the subcategory of ∂kUn(X, EX ) whose objects are k- ∗ linear ∂ -functors from (CX , EX ) to complete (i.e. having limits of small diagrams) k-linear categories CY and morphisms are pairs (F, φ) such that the functor F preserves limits. Dually, for a left exact additive k-linear category (CX, IX), we denote by ∂kUn(X, IX) the category whose objects are universal k-linear ∂-functors from (CX, IX) to k-linear 0 additive categories. Given two universal k-linear ∂-functors T and T from (CX, IX) to 0 respectively CY and CZ , a morphism from T to T is a pair (F, ψ), where F is a k- linear functor from CY to CZ preserving filtered colimits and ψ is a functor isomorphism T 0 −→∼ F ◦ T . The composition is defined by (F 0, ψ0) ◦ (F, ψ) = (F 0 ◦ F,F 0ψ ◦ ψ0). c We denote by ∂kUn (X, IX) the subcategory of ∂kUn(X, IX) whose objects are k-linear ∂-functors with values in cocomplete categories and morphisms are pairs (F, ψ) such that the functor F preserves colimits.

3.7.2.1. Proposition. Let (CX , EX ) be a svelte k-linear right exact category and ∗ ∗ (CX, IX) a svelte k-linear left exact category. The categories ∂kUn(X, EX ), ∂kUnc(X, EX ), c ∂kUn(X, IX), and ∂kUn (X, IX) have initial objects. Proof. The argument is similar to that of 3.7.1, except for we replace the category ∧ ∨ op CX (resp. CX) of presheaves of sets on CX (resp. on CX ) by the category Mk(X) (resp. o op Mk(X )) of presheaves of k-modules on CX (resp. on CX = CXo ). c (a) The initial object of the category ∂kUn (X, IX) is the universal k-linear ∂-functor Ext• from (C , I ) to the category M (X) of presheaves of k-modules on C whose X,IX X X k X zero component is the Yoneda embedding CX −→ Mk(X),L 7−→ CX(−,L). (b) The initial object of the category ∂ Un(X, I ) is the corestriction of Ext• to k X X,IX the smallest additive strictly full subcategory of Mk(X) which contains all presheaves Extn (L),L ∈ ObC , n ≥ 0. X,IX X (c) The universal k-linear ∂∗-functor Ext• from the right exact k-linear category (X,EX ) o op (CX , EX ) to the category Mk(X ) of presheaves of k-modules on CXo = CX is an initial ∗ object of the category ∂kUnc(X, EX ). (d) The corestriction of the ∂∗-functor Ext• to the smallest strictly full additive (X,EX ) subcategory of M (Xo) spanned by the presheaves Extn (L),L ∈ ObC , n ≥ 0, is k (X,EX ) X ∗ an initial object of the category ∂kUn(X, EX ). The argument is similar to that of 3.7.1. Details are left to the reader. 3.8. Universal problems for universal ’exact’ ∂∗- and ∂-functors. Fix a right ∗ exact category (CX , EX ) with an initial object. Let ∂ UEx(X, EX ) denote the category ∗ whose objects are universal ’exact’ ∂ -functors T = (Ti, di | i ≥ 0) from (CX , EX ) to ∗ right exact categories (CY , EY ) satisfying (CE5 ) (cf. 3.5.1) such that the functor T0 ∗ maps deflations to deflations. Let T be a universal ’exact’ ∂ - functor from (CX , EX ) to ∗ (CY , EY ) and Te a universal ’exact’ ∂ - functor from (CX , EX ) to (CZ , EZ ). A morphism 0 from T to T is a pair (F, φ), where F is a functor from CY to CZ which preserves filtered

38 limits and conﬂations, and φ is an isomorphism of ∂∗-functors F ◦ T −→∼ T 0. It (F 0, φ0) is a morphism from T 0 to T 00, then the composition of (F, φ) and (F 0, φ0) is deﬁned by (F 0, φ0) ◦ (F, φ) = (F 0 ◦ F, φ0 ◦ F 0φ). ∗ ∗ We denote by ∂ UExc(X, EX ) the subcategory of ∂ UEx(X, EX ) whose objects are ∗ ∗ ∂ -functors from (CX , EX ) to complete right exact categories (CY , EY ) satisfying (CE5 ) and morphisms are pairs (F, φ) such that the functor F preserves limits.

Dually, for a left exact category (CX, IX) with a final object, we denote by ∂UEx(X, IX) the category whose objects are universal ’exact’ ∂-functors T = (Ti, di | i ≥ 0) from (CX, IX) to left exact categories satisfying (CE5) such that the functor T0 maps inflations 0 to inflations. Given two universal ’exact’ ∂-functors T and T from (CX, IX) to respectively 0 (CY , IY ) and (CZ , IZ ), a morphism from T to T is a pair (F, ψ), where F is a functor from CY to CZ preserving filtered colimits and conflations and ψ is a functor isomorphism T 0 −→∼ F ◦ T . The composition is defined by (F 0, ψ0) ◦ (F, ψ) = (F 0 ◦ F,F 0ψ ◦ ψ0). c We denote by ∂UEx (X, IX) the subcategory of ∂UEx(X, IX) whose objects are ∂- functors with values in cocomplete left exact categories (with final objects) satisfying (CE5) and morphisms are pairs (F, ψ) such that the functor F preserves colimits.

3.8.1. Proposition. Let (CX , EX ) be a svelte right exact category with initial objects ∗ and (CX, IX) a svelte left exact category with ﬁnal objects. The categories ∂ UEx(X, EX ), ∗ c ∂ UExc(X, EX ), ∂UEx(X, IX), and ∂UEx (X, IX) have initial objects. Proof. (a) The Yoneda embedding

h X ∧ CX −−−→ CX,L 7−→ Lb = CX(−,L) is a fully faithful left exact functor. Therefore, it maps strict monomorphisms (in particular, ∧ inflations – arrows of IX) to strict monomorphisms of CX which happen to be universally ∧ ∧ strict. We denote by IX the coarsest left exact structure on CX which contains hX(IX ) and is closed with respect to inductive colimits. Since hX is a left exact functor, it is left ’exact’ functor from the left exact category ∧ ∧ (CX, IX) to the left exact category (CX, IX). Therefore, by (the dual version of) 3.6.1, the universal ∂-functor Ext• from (C , I ) to C∧ whose zero component is the Yoneda X,IX X X X ∧ ∧ embedding hX is an ’exact’ ∂-functor from (CX, IX) to (CX, IX). The claim is that the universal ’exact’ ∂-functor Ext• from (C , I ) to (C∧, I∧ ) X,IX X X X X c is an initial object of the category ∂UEx (X, IX). In fact, let (CZ , IZ ) be a left exact category such that the category CZ is cocomplete and F a left ’exact’ functor from (CX, IX) to (CZ , IZ ). Then the corresponding continuous ∗ ∧ F ∧ ∧ functor CX −→ CZ is an ’exact’ functor from (CX, IX) to (CZ , IZ ). Since the functor F ∗ is right exact, it suffices to show that F ∗ maps inflations to ∧ ∧ inflations, i.e. IX to IZ . The arrows of IX are obtained from the class of (strict) ∗ monomorphisms hX(IX) via compositions and push-forwards. The functor F preserves ∗ push-forwards and (any functor preserves) compositions. Since F = F ◦ hX, the class of ∗ morphisms F (hX(IX)) coincides with the class of monomorphisms F (IX). Therefore, it ∧ ∗ follows from the above description of IX (and the fact that F preserves push-forwards) ∗ ∧ that F (IX) is contained in IZ .

39 (b) The initial object of the category ∂UEx(X, IX) is the corestriction of the univer- • sal ∂-functor Ext to the smallest strictly full subcategory of C o which contains all X,IX XI sheaves Extn (L),L ∈ ObC , n ≥ 0. X,IX X (c) The universal ∂∗-functor Ext• from the right exact category (C , E ) to the (X,EX ) X X ∧ op category CXo of presheaves of sets on CXo = CX endowed with the coarsest right exact ∗ structure containing the image of EX is an initial object of the category ∂ UExc(X, EX ). (d) The corestriction of the ∂∗-functor Ext• to the smallest strictly full subcat- (X,EX ) ∧ n egory of C o spanned by the presheaves Ext (L),L ∈ ObC , n ≥ 0, is an initial X (X,EX ) X ∗ object of the category ∂ UEx(X, EX ). The argument is similar to that of 3.7.1. Details are left to the reader.

3.8.2. The k-linear version. Fix a right exact k-linear category (CX , EX ). Let ∗ ∗ ∂kUEx(X, EX ) denote the category whose objects are universal ’exact’ k-linear ∂ -functors T = (Ti, di | i ≥ 0) from (CX , EX ) to right exact k-linear categories (CY , EY ) satisfying ∗ (CE5 ) such that T0 maps deflations to deflations. Let T be a universal ’exact’ k-linear ∗ ∗ ∂ - functor from (CX , EX ) to (CY , EY ) and Te a universal ’exact’ k-linear ∂ - functor 0 from (CX , EX ) to (CZ , EZ ). A morphism from T to T is a pair (F, φ), where F is a k-linear functor from CY to CZ which preserves filtered limits and conflations, and φ is an isomorphism of ∂∗-functors F ◦ T −→∼ T 0. If (F 0, φ0) is a morphism from T 0 to T 00, then the composition of (F, φ) and (F 0, φ0) is defined by (F 0, φ0) ◦ (F, φ) = (F 0 ◦ F, φ0 ◦ F 0φ). ∗ ∗ We denote by ∂kUExc(X, EX ) the subcategory of ∂kUEx(X, EX ) whose objects are ∗ ∂ -functors from (CX , EX ) to complete right exact categories (CY , EY ) and morphisms are pairs (F, φ) such that the functor F preserves limits.

Dually, for a left exact k-linear category (CX, IX), we denote by ∂kUEx(X, IX) the category whose objects are universal ’exact’ k-linear ∂-functors T = (Ti, di | i ≥ 0) from (CX, IX) to k-linear left exact categories satisfying (CE5) such that the functor T0 maps inflations to inflations. Given two universal ’exact’ k-linear ∂-functors T and T 0 from 0 (CX, IX) to respectively (CY , IY ) and (CZ , IZ ), a morphism from T to T is a pair (F, ψ), where F is a k-linear functor from CY to CZ preserving filtered colimits and conflations and ψ is a functor isomorphism T 0 −→∼ F ◦ T . The composition is defined by (F 0, ψ0) ◦ (F, ψ) = (F 0 ◦ F,F 0ψ ◦ ψ0). c We denote by ∂kUEx (X, IX) the subcategory of ∂kUEx(X, IX) whose objects are k- linear ∂-functors with values in cocomplete left exact categories and morphisms are pairs (F, ψ) such that the functor F preserves colimits.

3.8.2.1. Proposition. Let (CX , EX ) be a svelte k-linear right exact category and ∗ (CX, IX) a svelte k-linear left exact category. The deﬁned above categories ∂kUEx(X, EX ), ∗ c ∂kUExc(X, EX ), ∂kUEx(X, IX), and ∂kUEx (X, IX) have initial objects. Proof. The argument is similar to that of 3.8.1, except for we replace the category ∧ ∨ op CX (resp. CX) of presheaves of sets on CX (resp. on CX ) by the category Mk(X) (resp. o op Mk(X )) of presheaves of k-modules on CX (resp. on CX = CXo ). (a) For a svelte k-linear left exact category (CX, IX), we denote by IX,k the coarsest left exact structure on the category Mk(X) of presheaves of k-modules on CX closed under hX,k inductive colimits and such that the Yoneda embedding CX −−−→ Mk(X) maps inﬂations

40 to inﬂations (i.e. IX to IX,k) and is a left exact k-linear functor, hence it is a left ’exact’ functor from (CX, IX) to (Mk(X), IX,k). Therefore, by the k-linear version of 3.6.1, the universal functor Ext• whose zero component is the Yoneda embedding h is ’exact’. X,IX X,k If (CZ , IZ ) be a left exact k-linear category such that the category CZ is cocomplete and F a left ’exact’ k-linear functor from (CX, IX) to (CZ , IZ ), then the corresponding ∗ ∧ F continuous functor CX −→ CZ is an ’exact’ functor from (Mk, IX,k) to (CZ , IZ ). The argument is similar to that of the corresponding part of 3.8.1. This implies that the universal k-linear ∂-functor Ext• from (C , I ) to the left X,IX X X c exact category (Mk(X), IX,k) is the initial object of the category ∂kUEx (X, IX). (b) The initial object of the category ∂ UEx(X, I ) is the corestriction of Ext• to k X X,IX the smallest additive strictly full left exact subcategory of (Mk(X), IX,k) which contains all presheaves Extn (L),L ∈ ObC , n ≥ 0. X,IX X (c) It follows from (a) (by duality) that the universal k-linear ∂∗-functor Ext• (X,EX ) o from the right exact k-linear category (CX , EX ) to the category Mk(X ) of presheaves op ∗ of k-modules on CXo = CX is an ’exact’ universal k-linear ∂ -functor from (CX , EX ) to o the right exact category (M (X ), E o ), where E o is the coarsest right exact structure k Xk Xk o o on M (X ) such that the Yoneda embedding C o −→ M (X ) maps E to E o . This k X k X Xk ∗ ∗ ’exact’ universal k-linear ∂ -functor is an initial object of the category ∂kUExc(X, EX ). (d) The corestriction of the ∂∗-functor Ext• to the smallest strictly full addi- (X,EX ) tive right exact subcategory of M (Xo) spanned by the presheaves Extn (L),L ∈ k (X,EX ) ∗ ObCX , n ≥ 0, is an initial object of the category ∂kUEx(X, EX ). Details are left to the reader.

3.9. Relative satellites. Fix a right exact category (CS, ES). Fix an object Y of CS and consider the right exact category (Y\CS, EY\S), where EY\S denote the right exact structure on Y\CS induced by ES.

∗ Y F Y 3.9.1. The ∂ -functor F• . For a functor CS −→ CZ , let F0 denote the composi- F tion of the canonical functor Y\CS −→ CS and CS −→ CZ . Suppose that the category CZ has initial objects, kernels of arrows, and limits of ﬁltered systems. Then the functor Y ∗ Y Y Y F0 extends to a (unique up to isomorphism) ∂ -functor F• = (Fn , dn | n ≥ 0) from the right exact category (Y\CS, EY\S) to CZ . If the category CS has initial objects and Y is Y one of them, then the category Y\CS is isomorphic to the category CS and the functor F• ∗ is the composition of this isomorphism and the universal ∂ -functor F•, where F0 = F. ξ It follows from the deﬁnition of satellites that for every object (V, Y −→V V) of the category Y\CS, we have

p Y Y W F1 (V, ξV ) = SY (F)(V, ξV ) = lim Ker(F(Y W −→ W)), (1) e,ξV

where pW is the canonical projection and the limit is taken by the ﬁltered system of e Y n deﬂations (W, ξW ) −→ (V, ξV ). By (the argument of) 3.3.2, Fn = SY (F) for all n ≥ 0.

∗ Y,ES 3.9.2. The ∂ -functor F• . Let CZ be a category with ﬁnal objects and cok- F ernels of arbitrary morphisms. For any functor CS −→ CZ , let FY denote the func-

41 ξ tor Y\CS −→ CZ which assigns to every object (W, Y → W) the object Cok(F (ξ)) and acts correspondingly on morphisms. Notice that the functor FY maps the initial object (Y, idY ) of the category Y\CS to a ﬁnal object of the category CZ . If, in addition, the category CZ has initial objects (e.g. it is pointed), kernels of arrows and limits of ﬁltered systems, then there exists a (unique up to isomorphism) universal ∂∗-functor Y,ES Y,ES Y,ES Y,ES F• = (Fn , dn | n ≥ 0) such that F0 = FY . 4. Stable categories of left exact categories. Suspended categories and cohomological functors.

4.1. Observations. Let (CX , IX ) be a svelte left exact category with a final object x and CY a category with a final object y and arbitrary colimits. Then, by the (dual version of the) argument of 3.3.2, we have an endofunctor S+ of the category Hom(CX ,CY ) λ of functors from CX to CY , together with a cone y −→ S+, where y is the constant functor with the values in the final object y of the category CY . For any conflation j e F E = (N −→ M −→ L) of (CX , IX ) and any functor CX −→ CY , we have a commutative diagram F j F e F (N) −−−→ F (M) −−−→ F (L)     y y d0(E) (1) λ(N) y −−−→ S+F (N)

0 0 e Here S+F (N) = colim(Cok(F (M −→ N))), where the colimit is taken by the diagram 0 0 of all conﬂations N −→j M 0 −→e L (see the argument of 3.3.2). By [GZ, II.1.3], there is a natural equivalence between the category Hom(CX ,CY ) and the category of functors from CX and CY and the category Homc(CX ,CY ) of continuous (i.e. having a right adjoint, or, ∧ ∗ equivalently, preserving colimits) functors from CX to CY . Let F denote a (determined ∗ uniquely up to isomorphism) continuous functor corresponding to F , i.e. F = F ◦ hX , ∧ where hX is the Yoneda embedding CX −→ CX ,M 7−→ Mc = CX (−,M). Since the ∗ functor F preserves colimits, the formula for S+F (N) can be rewritten as follows:

0 0 0 e ∗ 0 e S+F (N) = colim(Cok(F (M −→ N))) = colim(Cok(F (Mc −→b Nb))) (2) 0 ∗ 0 e ∗ ∗ 1 = F (colim(Cok(Mc −→b Nb))) = F S+hX (N) = F ExtX (N).

0 0 where colimit is taken by the diagram of all conﬂations N −→j M 0 −→e L. For any presheaf of sets G on CX , we set

∧ 1 Θb X∗(G)(−) = CX (ExtX (−), G). (3)

Θ ∧ bX∗ ∧ The map G 7−→ Θb X∗(G) extends to an endofunctor CX −−−→ CX . It follows from the deﬁnition of Θb X∗ (and the Yoneda’s formula) that

∧ 1 ∧ CX (ExtX (−), G) = Θb X∗(G)(−) ' CX (−, Θb X∗(G)). (4)

42 1∗ ∧ ∧ 1 Let ExtX denote the continuous functor CX −→ CX corresponding to ExtX . It ∧ 1∗ ∧ follows from (4) that CX (ExtX (−), G) ' CX (−, Θb X∗(G)), that is the functor Θb X∗ is a 1∗ ∗ 1∗ right adjoint to ExtX . For convenience, we shall write Θb X instead of ExtX . Taking as F the Yoneda functor hX (and setting Nb = hX (N)), we obtain from the diagram (1) the diagram

j e N −−−→b M −−−→b L b c b   y y d0(E) (5) λ(N) b ∗ xb −−−→ Θb X (Nb)

∗ Once the functor Θb X is given, all the information about the universal ∂-functor • i • ExtX = (ExtX , di | i ≥ 0), and, therefore, due to the universality of ExtX , all the information about all universal ∂-functors from the left exact category (CX , IX ), is contained in the diagrams (5), where N −→j M −→e L runs through the class of conﬂations of (CX , IX ). • ∗n ∗n In fact, the universal ∂-functor ExtX is of the form (Θb X ◦ hX , Θb X (d0)|n ≥ 0); and for any functor F from CX to a category CY with colimits and ﬁnal objects, the universal ∂-functor (Ti, di | i ≥ 0) from (CX , IX ) to CY with T0 = F is isomorphic to

∗ • ∗ ∗n ∗ ∗n F ◦ ExtX = (F Θb X ◦ hX ,F Θb X (d0) | n ≥ 0). (6)

4.1.1. Note. If CX is a pointed category, then the presheaf xb = CX (−, x) is both a λ(N) ∧ b ∗ ﬁnal and an initial object of the category CX . In particular, the morphism xb −−−→ Θb X (Nb) in (5) is uniquely deﬁned, hence is not a part of the data. In this case, the data consists of diagrams

j e d0(E) b b ∗ Nb −−−→ Mc −−−→ Lb −−−→ Θb X (Nb),

j e where E = (N −→ M −→ L) runs through conﬂations of (CX , IX ). 4.2. Stable category of a left exact category. Consider the full subcategory ∧ ∗n CXp of the category CX whose objects are Θb X (M), where M runs through representable presheaves and n through nonnegative integers. We denote by θXp the endofunctor ∗ ∗ CXp −→ CXp induced by Θb X . It follows that CXp is the smallest Θb X -stable strictly full ∧ subcategory of the category CX containing all presheaves Mc = CX (−,M),M ∈ ObCX . 4.2.1. Triangles. We call the diagram

j e d0(E) b b ∗ Nb −−−→ Mc −−−→ Lb −−−→ Θb X (Nb), (1)

j e where E = (N −→ M −→ L) is a conﬂation in (CX , IX ), a standard triangle.

43 A triangle is any diagram in CXp of the form

j e d N −→ M −→ L −→ θXp (N ), (2) which is isomorphic to a standard triangle. It follows that for every triangle, the diagram

e M −−−→ L     y y d λ(N ) ∗ xb −−−→ Θb X (N ) commutes. Triangles form a category TrXp : morphisms from

j e d N −−−→ M −−−→ L −−−→ θX(N ) to j0 e0 d0 0 0 0 0 N −−−→ M −−−→ L −−−→ θX(N ) are given by commutative diagrams

j e d N −−−→ M −−−→ L −−−→ θX(N )         f y y g y h y θX(f) j0 e0 d0 0 0 0 0 N −−−→ M −−−→ L −−−→ θX(N ) The composition is obvious. 4.2.2. The prestable category of a left exact category. Thus, we have obtained a data (CXp , (θXp , λ), TrXp ). We call this data the prestable category of the left exact category (CX , IX ). 4.2.3. The stable category of a left exact category with ﬁnal objects. Let

(CX , IX ) be a left exact category with a ﬁnal object x and (CXp , θXp , λ; TrXp ) the associated with (C , I ) presuspended category. Let Σ = Σ be the class of all arrows t of X X θXp

CXp such that θXp (t) is an isomorphism. −1 We call the quotient category CXs = Σ CXp the stable category of the left exact category (CX , IX ). The endofunctor θXp determines a conservative endofunctor θXs of q∗ Σ the stable category CXs . The localization functor CXp −→ CXs maps ﬁnal objects to ﬁnal objects. Let λ denote the image x = q∗ (x) −→ θ of the cone x −→λ θ . s e Σ b Xs b Xp Finally, we denote by TrXs the strictly full subcategory of the category of diagrams of the form N −→ M −→ L −→ θ (N ) generated by q∗ (Tr ). Xs Σ Xs

The data (CXs , θXs , λs; TrXs ) will be called the stable category of (CX , IX ).

4.2.4. Dual notions. If (CX, EX) is a right exact category with an initial object, one obtains, dualizing the deﬁnitions of 4.2.2 and 4.4.3, the notions of the precostable and costable category of (CX, EX).

44 4.3. Presuspended categories. Fix a category CX with a ﬁnal object x and a eθX functor CX −−−→ x\CX, or, what is the same, a pair (θX, λ), where θX is an endofunctor CX −→ CX and λ is a cone x −→ θX. We denote by TrfX the category whose objects are all diagrams of the form

j e d N −−−→ M −−−→ L −−−→ θX(N ) such that the square e M −−−→ L     y y d λ(N ) x −−−→ θX(N ) commutes. Morphisms from

j e d N −−−→ M −−−→ L −−−→ θX(N ) to j0 e0 d0 0 0 0 0 N −−−→ M −−−→ L −−−→ θX(N )

f g are triples (N −→ N 0, M −→ M0, L −→h L0) such that the diagram

j e d N −−−→ M −−−→ L −−−→ θX(N )         f y y g y h y θX(f) j0 e0 d0 0 0 0 0 N −−−→ M −−−→ L −−−→ θX(N ) commutes. The composition of morphisms is natural.

4.3.1. Deﬁnition. A presuspended category is a triple (CX, θeX, TrX), where CX and θeX = (θX, λ) are as above and TrX is a strictly full subcategory of the category TrfX whose objects are called triangles, which satisﬁes the following conditions:

(PS1) Let CX0 denote the full subcategory of CX generated by objects N such that j e d there exists a triangle N −→ M −→ L −→ θX(N ). For every N ∈ ObCX0 , the diagram

idN λ(N ) N −−−→ N −−−→ x −−−→ θX(N ) is a triangle. j e d f 0 (PS2) For any triangle N −→ M −→ L −→ θX(N ) and any morphism N −→ N 0 0 0 0 0 j 0 e 0 d 0 with N ∈ ObCX0 , there is a triangle N −→ M −→ L −→ θX(N ) such that f extends to a morphism of triangles

(f,g,h) 0 0 0 j e d 0 j 0 e 0 d 0 (N −→ M −→ L −→ θX(N )) −−−−−−−→ (N −→ M −→ L −→ θX(N )).

45 (PS3) For any pair of triangles

0 0 0 j e d 0 j 0 e 0 d 0 N −→ M −→ L −→ θX(N ) and N −→ M −→ L −→ θX(N ) and any commutative square j N −−−→ M     f y y g j0 N 0 −−−→ M0 there exists a morphism L −→h L0 such that (f, g, h) is a morphism of triangles, i.e. the diagram j e d N −−−→ M −−−→ L −−−→ θX(N )         f y y g y h y θX(f) j0 e0 d0 0 0 0 0 N −−−→ M −−−→ L −−−→ θX(N ) commutes. (PS4) For any pair of triangles

u v w x 0 s r N −→ M −→ L −→ θX(N ) and M −→ M −→ Mf −→ θX(M), there exists a commutative diagram

u v w N −−−→ M −−−→ L −−−→ θX(N )         id y x y y y y id u0 v0 w0 0 0 N −−−→ M −−−→ L −−−→ θX(N )       s y y t y θX(u) (2) id r Mf −−−→ Mf −−−→ θX(M)     0 r y y r θX(v) θX(M) −−−→ θX(L) whose two upper rows and two central columns are triangles. j e d (PS5) For every triangle N −→ M −→ L −→ θX(N ), the sequence

... −−−→ CX(θX(N ), −) −−−→ CX(L, −) −−−→ CX(M, −) −−−→ CX(N , −) is exact.

4.3.1.1. Remarks. (a) If CX is an additive category, then three of the axioms above coincide with the corresponding Verdier’s axioms of triangulated category (under condition

46 that CX0 = CX). Namely, (PS1) coincides with the ﬁrst half of the axiom (TRI), the axiom (PS3) coincides with the axiom (TRIII), and (PS4) with (TRIV) (see [Ve2, Ch.II]). (b) It follows from (PS4) that if N −→ M −→ L −→ θX(N ) is a triangle, then all three objects, N , M, and L, belong to the subcategory CX0 . (c) The axiom (PS2) can be regarded as a base change property, and axiom (PS4) expresses the stability of triangles under composition. So that the axioms (PS1), (PS2) and (PS4) say that triangles form a ’pretopology’ on the subcategory CX0 . The axiom (PS5) says that this pretopology is subcanonical: the representable presheaves are sheaves. These interpretations (as well as the axioms themselves) come from the main examples: prestable and stable categories of a left exact category.

4.3.2. The category of presuspended categories. Let T+CX = (CX, θX, λX; T rX) and T+CY = (CY, θY, λY; T rY) be presuspended categories. A triangle functor from T+CX to T+CY is a pair (F, φ), where F is a functor CX −→ CY which maps initial object to an initial object and φ is a functor isomorphism F ◦ θX −→ θY ◦ F such that for every u v w triangle N −→ M −→ L −→ θX(N ) of T+CX, the sequence

F (u) F (v) φ(N )F (w) F (N ) −−−→ F (M) −−−→ F (L) −−−→ θY(F (N )) is a triangle of T+CY. The composition of triangle functors is deﬁned naturally:

(G, ψ) ◦ (F, φ) = (G ◦ F, ψF ◦ Gφ).

0 0 Let (F, φ) and (F , φ ) be triangle functors from T−CX to T−CY. A morphism from (F, φ) to (F 0, φ0) is given by a functor morphism F −→λ F 0 such that the diagram

φ θY ◦ F −−−→ F ◦ θX     θYλ y y λθX φ0 0 0 θY ◦ F −−−→ F ◦ θX commutes. The composition is the compsition of the functor morphisms. Altogether gives the deﬁnition of a bicategory PCat formed by svelte presuspended categories, triangle functors as 1-morphisms and morphisms between them as 2-morphisms. As usual, the term “category PCat” means that we forget about 2-morphisms. Dualizing (i.e. inverting all arrows in the constructions above), we obtain the bicategory PoCat formed by precosuspended svelte categories as objects, triangular functors as 1-morphisms, and morphisms between them as 2-morphisms.

4.4. Quasi-suspended categories. We call a presuspended category (CX, θX, λ; TrX) quasi-suspended if the functor θX is conservative. We denote by SCat the full subcategory of the category PCat whose objects are quasi-suspended svelte categories.

Let (CX, θX, λ; TrX) be a presuspended category and Σ = ΣθX the class of all arrows s of the category CX such that θX(s) is an isomorphism. Let ΘX denote the endofunctor −1 ∗ ∗ of the quotient category Σ CX uniquely determined by the equality ΘX ◦ qΣ = qΣ ◦ θX,

47 ∗ −1 ∗ where qΣ denotes the localization functor CX −→ Σ CX. Notice that the functor qΣ ∗ maps ﬁnal objects to ﬁnal objects. Let λe denote the morphism qΣ(x) −→ ΘX induced by q∗ (λ) λ ∗ Σ ∗ ∗ x −→ θX (that is by qΣ(x) −−−→ qΣ ◦ θX = ΘX ◦ qΣ) and TrfX the essential image of TrX. −1 Then the data (Σ CX, ΘX, λe; TrfX) is a quasi-suspended category. The constructed above map

−1 (CX, θX, λ; TrX) 7−→ (Σ CX, ΘX, λe; TrfX)

∗ extends to a functor PCat −→J SCat which is a left adjoint to the inclusion functor SCat −→J∗ PCat. The natural triangle (localization) functors

q∗ Σ −1 (CX, θX, λ; TrX) −−−→ (Σ CX, ΘX, λe; TrfX)

∗ form an adjunction arrow IdPCat −→ J∗J . The other adjunction arrow is identical. 4.5. The stable category of a left exact category with ﬁnal objects. Let

(CX , IX ) be a left exact category with a ﬁnal object x and (CXp , θXp , λ; TrXp ) the as- −1 sociated with (CX , IX ) presuspended category. We call the category Σ CXp the stable category of the left exact category (CX , IX ). The corresponding quasi-suspended category −1 (Σ CXp , ΘXp , λe; TrfXp ) will be called the stable quasi-suspended category of (CX , IX ).

4.5.1. Proposition. Let (CX , IX ) be a left exact category with ﬁnal objects. Suppose that (CX , IX ) has enough pointed (i.e. having a morphism from a ﬁnal object) injectives. Then the stable quasi-suspended category of (CX , IX ) is naturally equivalent to its weak stable category.

−1 Proof. It is easy to see that the natural functor CX −→ Σ CX factors through the weak stable category of (CX , IX ). The claim is that the corresponding (unique) functor −1 from the weak stable category of (CX , IX ) to Σ CX is a category equivalence. 4.6. Homology and homotopy of ’spaces’. 4.6.1. Homology of ’spaces’ with coefficients in a right exact category. Let CX be a svelte category and (CZ , EZ ) a svelte right exact category with colimits and initial objects. We denote by CH(Z,X) the category of functors CX −→ CZ . F We define the zero homology group of a ’space’ X with coefficients in CX −→ CZ by H0(X, F) = colim F. The higher homology groups, Hn(X, F), are values at F of satellites H0(X,−) of the functor CH(Z,X) −−−→ CZ with respect to the (object-wise) right exact structure EH(Z,X) induced by EZ (cf. K.2.6). Thus, we have a universal ∂∗-functor

H•(X, −) = (Hn(X, −), dn | n ≥ 0) from the right exact category of coeﬃcients (CH(Z,X), EH(Z,X)) to (CZ , EZ ).

48 4.6.1.1. Proposition. Suppose that the right exact category (CZ , EZ ) satisﬁes ∗ ∗ (CE5 ). Then the universal ∂ -functor H•(X, −) is ’exact’.

Proof. Let J∗ denote the canonical embedding of the category CZ into the category CH(Z,X) = Hom(CX ,CZ ) which assigns to every object M of the category CZ the con- ∗ stant functor mapping all arrows of CX to idM . The functor J∗ has a left adjoint, J , which assigns to every functor CX −→ CZ its colimit and to every functor morphism the ∗ corresponding morphism of colimits. The composition J J∗ is (isomorphic to) the identi- ∗ cal functor; i.e. J is a (continuous) localization functor. The functor J∗ is exact, hence ∗ ’exact’, for any category CX . The functor J is right exact (in particular right ’exact’), because it has a right adjoint. The assertion follows now from 3.6.1. 4.6.1.2. Note. There is a natural equivalence between the category of local systems of abelian groups on the classifying topological space B(X) of a category CX and the morphism inverting functors from CX to Z − mod. If F is a morphism inverting functor CX −→ Z − mod and LF the corresponding local system, then the homology groups Hn(X, F) are naturally isomorphic to the homology groups Hn(B(X), LF ) of the classifying space B(X) with coeﬃcients in the local coeﬃcient system LF (cf. [Q, Section 1].

4.6.2. The ’space’ of paths of a ’space’. Let Pa∗ be the functor from Cat to the category of diagrams of sets of the form A ⇒ B which assigns to each category CX s −→ the diagram HomCX −→ObCX , where s maps an arrow to its source and t to its target. r ∗ The functor Pa∗ has a left adjoint, Pa , which assigns to each diagram T = (T1 ⇒ T0) ε(X)∗ ∗ ∗ the category Pa (T ) of paths of T . The adjunction morphism Pa Pa∗(CX ) −−−→ CX is a functor which is identical on objects and mapping each path of arrows

M1 −→ M2 −→ ... −→ Mn to its composition M1 −→ Mn. ∗ We denote by Pa(X) the ’space’ represented by the category Pa Pa∗(CX ) and call it the ’space’ of paths of the ’space’ X. The map X 7−→ Pa(X) extends to an endo- ε(X)∗ o ∗ functor, Pa, of the category |Cat| . The adjunction morphism Pa Pa∗(CX ) −−−→ CX is ε(X) interpreted as an inverse image functor of a morphism of ’spaces’ X −−−→ Pa(X). The o morphisms ε = (ε(X) | X ∈ Ob|Cat| ) form a functor morphism Id|Cat|o −→ Pa. 4.6.2.1. The ’space’ of paths and the loop ’space’ of a pointed ’space’. Consider the pointed category |Cat|o/x associated with the category of ’spaces’ |Cat|o; Here x is the initial object of |Cat|o represented by the category with one (identical) morphism. By C1.5, a choice of a pseudo-functor

c o op ∗ ∗ f,g ∗ ∗ |Cat| −−−→ Cat ,X 7−→ CX , f 7−→ f ;(gf) −−−→ f g ,

o o induces an equivalence between the category |Cat| /x and the category |Cat|x whose objects are pairs (X, OX ), where OX ∈ ObCX ; morphisms from (X, OX ) to (Y, OY )

49 are pairs (f, φ), where f is a morphism of ’spaces’ X −→ Y and φ is an isomorphism (f,φ) (g,ψ) ∗ f (OY ) −→ OX . The composition of (X, OX ) −−−→ (Y, OY ) −−−→ (Z, OZ ) is the mor- ∗ phism (g ◦ f, φ ◦ f (ψ) ◦ cf,g). o o The endofunctor Pa of |Cat| induces an endofunctor Pax of |Cat|x which assigns to each pointed ’space’ (X, OX ) the pointed ’space’ (Pa(X), OX ) of paths of (X, OX ). It ε(X) follows that the canonical morphism X −−−→ Pa(X) is a morphism of pointed ’spaces’ (X, OX ) −→ Pax(X, OX ) = (Pa(X), OX ). It follows from C1.5.1 that the category representing the cokernel of the canonical ε(x) morphism (X, OX ) −−−→ Pax(X, OX ) is the subcategory of the category CPa(X) whose objects are isomorphic to OX and morphisms are paths of arrows M1 −→ ... −→ Mn whose composition is an isomorphism. This category is equivalent to its full subcategory

CΩ(X,OX ) of CPa(X) which has one object, OX . We call the ’space’ Ω(X, OX ) represented by the latter category the loop ’space’ of the pointed ’space’ (X, OX ). spl 4.6.2.2. Left exact structures on the category of pointed ’spaces’. Let Ex −1 spl be the class of all split epimorphisms of diagrams A ⇒ B. By 2.6.3.2, the class Pa∗ (Ex ) is a right exact structure on the category of swelte pointed categories. This right exact o structure determines a left exact structure J0 on the category |Cat|x of pointed ’spaces’, o so that (|Cat|x, J0) is a Karoubian left exact category. Each path ’space’ (Pa(X), OX ) is o ε(X) an injective object of (|Cat|x, J0), and the canonical morphism (X, OX ) −→ (Pa(X), OX ) belongs to J0. The fact that every epimorphism of diagrams of the form A ⇒ B splits j implies that the class J0 consists of all morphisms (X, OX ) −→ (Y, OY ) of the pointed ∗ ’spaces’ such that the image of j is naturally equivalent to the category CX . 4.6.3. The ﬁrst homotopy group of a pointed ’space’. Given a svelte category CX , we denote by CG(X) the groupoid obtained from CX by localization at Hom(CX ). The o map G which assigns to each object (X, OX ) of the category |Cat|x the pair (G(X), OX ) (we identify objects of CG(X) with objects of CX ) is naturally extended to a functor from o o |Cat|x to its full subcategory Grx generated by objects (Y, OY ) such that CY is a groupoid. o o This functor is a left adjoint to the inclusion functor Grx −→ |Cat|x.

4.6.3.1. Definition. The fundamental group π1(X, OX ) of the pointed ’space’ (X, OX ) is the group CG(X)(OX , OX ) of the automorphisms of the object OX of the groupoid CG(X) associated with the category CX . (see [GZ, II.6.2]). By [Q, Proposition 1], the group π1(X, OX ) is isomorphic to the fundamental group π1(B(X), OX ) of the pointed classifying space (B(X), OX ) of the category CX . 4.6.4. Higher homotopy groups of a pointed ’space’. The map which assigns to every pointed ’space; (X, OX ) its fundamental group π1(X, OX ) is a functor from o op (|Cat|x) to the category Z − mod of abelian groups. The functor π1 maps every inflation to an epimorphism and every conflation (X, OX ) −→ (Y, OY ) −→ (Z, OZ ) to an exact sequence of abelian groups π1(Z, OZ ) −→ π1(Y, OY ) −→ π1(X, OX ). Therefore, by 3.3.2, ∗ o op the universal ∂ -functor π• = (πn, dn | n ≥ 1) from (|Cat|x, J0) to Z − mod is ’exact’. We call πn(X, OX ) the n-th homotopy group of the pointed ’space’ (X, OX ).

50 4.6.4.1. Proposition. For any pointed ’space’ (X, OX ) and any n ≥ 1, there is a natural isomorphism πn+1(X, OX ) ' πn(Ω(X, OX )). Proof. This follows from the long exact sequence

... −−−→ πn+1(Pa(X, OX )) πn(Pa(X, OX )) −−−→ ...  x   y  πn+1(X, OX ) −−−→ πn(Ω(X, OX )) corresponding to the (functorial) conﬂation (X, OX ) −→ Pa(X, OX ) −→ Ω(X, OX ) of pointed ’spaces’ and the fact that the ’space’ (Pa(X), OX ) is an injective object of the o op pointed left exact category (|Cat|x, J0) , hence πn(Pa(X), OX ) = 0 for n ≥ 1. 5. Projectives and injectives.

Fix a right exact category (CX , EX ).

5.1. Lemma. The following conditions on an object P of CX are equivalent: (a) Every deﬂation M −→ P splits. f (b) For every deﬂation M −→e N and a morphism P −→ N, there exists a morphism g P −→ M such that f = e ◦ g.

Proof. Obviously, (b) ⇒ (a): it suﬃces to take f = idP . (a) ⇒ (b). Since deﬂations are stable under any base change, there is a cartesian square f 0 M −−−→ M f  0   e y cart y e f P −−−→ N

g whose left vertical arrow is a deﬂation. By (a), it splits; i.e. there is a morphism P −→ Mf 0 0 0 0 such that e ◦ g = idP . Therefore, e ◦ (f ◦ g) = (e ◦ f ) ◦ g = (f ◦ e ) ◦ g = f.

5.2. Projectives. Let (CX , EX ) be a right exact category. We call an object P of CX a projective object of (CX , EX ), if it satisﬁes the equivalent conditions of 5.1. We denote by PEX the full subcategory of CX generated by projective objects.

5.2.1. Example. Let (CX , EX ) be a right exact category whose deﬂations are split. Then every object of CX is a projective object of (CX , EX ).

5.3. Right exact categories with enough projectives. We say that (CX , EX ) has enough projectives if for every object N of CX there exists a deﬂation P −→ N, where P is a projective object.

5.3.1. Proposition. Let (CX , EX ) and (CY , EY ) be right exact categories, and let f ∗ CY −→ CX be a functor having a right adjoint, f∗. If the functor f∗ maps deﬂations to deﬂations (e.g. it is right weakly ’exact’), then f ∗ maps projectives to projectives.

51 e ∗ Proof. Let P be a projective object of (CY , EY ) and M −→ f (P ) a deflation. Then, f (e) ∗ ∗ by hypothesis, f∗(M) −−−→ f∗f (P ) is a deflation. Since P is a projective object, there exists an arrow P −→t f ∗(M) such that the diagram P t η(P ) .& (1) f (e) ∗ ∗ f∗(M) −−−→ f∗f (P ) 0 commutes (here η(P ) is an adjunction arrow). Then the composition f ∗(P ) −→t M of f ∗(t) ε(M) ∗ ∗ ∗ f (P ) −−−→ f f∗(M) and the adjunction morphism f f∗(M) −−−→ M splits the deflation M −→e f ∗(P ). This follows from the commutativity of the diagram f ∗(P ) ∗ ∗ f (t) .&f η(P ) f ∗f (e) ∗ ∗ ∗ ∗ f f∗(M) −−−−−−−→ f f∗f (P ) (2)     ∗ ε(M) y y εf (P ) e M −−−−−−−→ f ∗(P ) ∗ ∗ and the equality εf ◦ f η = Idf ∗ .

5.3.2. Proposition. Let (CX , EX ) and (CY , EY ) be right exact categories, and let f ∗ CY −→ CX be a functor having a right adjoint, f∗. Suppose that EY consists of all split epimorphisms of CY and the functor f∗ maps deflations to deflations (that is split epimorphisms) and reflects deflations (i.e. if f∗(t) is a split epimorphism, then t is a deflation). Then (CX , EX ) has enough projectives.

Proof. Since EY consists of split epimorphisms, all objects of CY are projective. ∗ Therefore, by 5.3.1, every object of the form f (N),N ∈ ObCY , is projective. For every ε(M) ∗ object M of CX the adjunction morphism f f∗(M) −−−→ M is a deﬂation, because the morphism f∗ε(M) is a split epimorphism, hence, by hypothesis, it belongs to EY .

5.4. Right exact structure with a given class of projectives. Let CX be a f category and P a class of objects of CX . Let E(P) denote the class of all arrows M −→ L CX (f,P ) g of CX such that CX (P,M) −−−→ CX (P,L) is surjective and for any morphism N −→ L, there exists a pull-back of f along g. 5.4.1. Lemma. The class E(P) is the class of covers of a Grothendieck pretopology. Proof. Obviously, the class E(P) contains all isomorphisms and is closed under compositions. By assumption, for any morphism M −→t L of E(P) and an arbitrary morphism g N −→ L of CX , there exists a cartesian square g0 M −−−→ M f    et y cart y t (3) g N −−−→ L

52 The functor CX (P, −) preserves cartesian squares for any object P of CX . In particular, the image g0 M −−−→ M f    et y cart y t (4) g N −−−→ L of (3) is a cartesian square. If P belongs to P, then its right vertical of (4) is surjective, t hence its left vertical arrow is surjective too. Tbis shows the pull-back Mf −→e N of the morphism t belongs to E(P).

5.4.2. Proposition. Let CX be a category. For any class of objects P of the st def st T category CX , the class of morphisms EX (P) = EX E(P) is the ﬁnest among the right exact structures EX on CX such that all objects of P are projectives of (CX , EX ). st Proof. Recall that EX is the ﬁnest right exact structure on CX ; it consists of all universal strict epimorphisms of CX . The intersection of Grothendieck pretopologies is st a Grothendieck pretopology. Since it is contained in EX , it is a left exact structure. Evidently, any right exact structure EX such that all objects of P are projectives of st (CX , EX ), is coarser than EX (P).

5.5. Right exact categories of modules over monads. Fix a category CY such spl that the class EY of split epimorphisms of CY is stable under base change. Equivalently, f for each split epimorphism M −→t L and for an arbitrary morphism N −→ L of the category CY , there exists a cartesian square f 0 M −−−→ M f    et y cart y t f N −−−→ L

spl (whose left vertical arrow is split, because t is split). So that (CY , EY ) is a right exact category. Let F = (F, µ) be a monad on the category CY . Set CX = F − mod (i.e. X = Sp(F/Y ) – the spectrum of the monad F) and denote by f∗ the forgetful functor −1 spl CX −→ CY . We set EX = f∗ (EY ). Since f∗ preserves and reﬂects limits (in particular, pullbacks), the arrows of EX are covers of a pretopology, i.e. (CX , EX ) is a right exact f ∗ category. The functor f∗ has a left adjoint, V 7−→ (F (V ), µ(V )), and all together satisfy the conditions of 5.3.2. Therefore, (CX , EX ) has enough projectives. Explicitely, it follows from ∗ (the argument of) 5.3.2 that objects f (V ) = (F (V ), µ(V )) are projectives of (CX , EX ) ξ for all V ∈ ObCY , and for every F-module M = (M, ξ), the action F (M) −→ M can be regarded as a canonical deﬂation from a projective object:

∗ ξ f f∗(M) = (F (M), µ(M)) −→ M.

spl 5.5.1. Proposition. Suppose that (CY , EY ) is a Karoubian right exact category (i.e. CY is a Karoubian category and split epimorphisms are stable under base change).

53 Then for every monad F = (F, µ) on CY , the right exact category (F − mod, EX ), where spl EX is the induced by EY right exact structure, is Karoubian.

f∗ Proof. (a) The forgetful functor F − mod −→ CY reflects and preserves limits; in particular, it reflects and preserves pull-backs. Therefore, the stability of split epimorphisms of CY under base change implies the same property of split epimorphisms of F − mod. (b) It remains to show that F − mod is a Karoubian category. Let M = (M, ξ) be an F-module and p an idempotent M −→ M. Since CY is a Karoubian category, f∗(p) the idempotent f∗(M) = M −−−→ M splits. By 1.5.1, the latter is equivalent to the id −→M existence of the kernel of the pair of arrows M −→ M. Since the forgetful functor f∗ f∗(p) reflects and preserves limits, in particular kernels of pairs of arrows, there exists the kernel id −→M of pair of arrows M−→M; i.e. the idempotent p splits. p

5.5.2. Corollary. Let G = (G, δ) be a comonad on a Karoubian category CX . spl Suppose that class IX of split monomorphisms in CX is stable under cobase change (i.e. spl spl IX is a left exact structure on CX ). Let IY be the preimage of IX in the category CY = G −comod of G-comodules. Then (CY, IY) is a Karoubian left exact category having enough injectives. Proof. The assertion is dual to that of 5.5.1. Futher on, we need details which are as follows. Let CY be the category G − comod of G-comodules with the exact structure induced by the forgetful functor

g∗ CY = G − comod −−−→ CX .

Its right adjoint

g∗ CX −−−→ CY = G − comod, M 7−→ (G(M), δ(M)), (1) maps every object M of the category CX to an EY-injective object. If the category CX is Karoubian, then, for every object M = (M,M −→ν G(M)), the adjunction morphism

ν ∗ M −−−→ g∗g (M) = (G(M), δ(M)) (2) is an inﬂation of G-comodules (see the argument of the dual assertion 5.5.1). 5.5.3. Corollary. Under the conditions of 5.5.2, an object M = (M, ν) of the cate- ν gory CY of G-comodules is IY-injective iﬀ the adjunction morphism M −→ (G(M), δ(M)) splits (as a morphism of G-comodules).

5.5.4. Proposition. Suppose that CX is a Karoubian category whose split epimorphisms (resp. split monomorphisms) are stable under base (resp. cobase) change. Let F = (F, µ) be a continuous monad on CX (i.e. the functor F has a right adjoint) and

54 −1 spl f∗ the forgetful functor F − mod −→ CX . Set CX = F − mod, EX = f∗ (EX ), and −1 spl IX = f∗ (IX ). Then (CX, EX) is a right exact category with enough projectives and (CX, IX) is a left exact category with enough injectives. Proof. If the monad F = (F, µ) is continuous, i.e. the functor F has a right adjoint, ! f∗ F , then (and only then) the forgetful functor F − mod = CX −→ CX has a right adjoint, ! ! ! ! ! f , such that F = f∗f . Thus, we have the comonad F = (F , δ) corresponding to the ! pair of adjoint functors f∗, f and an isomorphism of categories

Φ F − mod −−−→ F ! − comod

ξ ξ which assigns to every F-module (M,F (M) −→ M) the F !-comodule (M,M −→b F !(M)) determined (uniquely up to isomorphism) by adjunction. It follows that the diagram

Φ F − mod −−−→ F ! − comod ∗ f∗ &. g (3) CX commutes. By 5.5.1, the category CX = F − mod has enough EX-injectives. By 5.5.2, ! the category CY = F − comod has enough IY-injectives. The functor Φ in (3) is an isomorphism of exact categories, hence the assertion.

5.6. Coeffaceable functors, universal ∂∗-functors, and projectives. Let (CX , EX ) be a right exact category and CY a category with an initial object. We call F a functor CX −→ CY coeffaceable, or EX -coeffaceable, if for any object L of CX , there exists a deflation M −→t L such that F (t) is a trivial morphism.

F 5.6.1. Coeffaceble functors and projectives. If a functor CX −→ CY is EX - coeffaceable, then the morphism F (t) is trivial for any projective deflation t, and F maps every projective object of (CX , EX ) to an initial object of CY . In fact, a projective deflation M −→t L factors through any other deflation of L; and, by hypothesis, there exists a deflation M −→e L such that F (e) is trivial. Therefore, the morphism F (t) is trivial. An object M is projective iff idM is a projective deflation; and the triviality of F (idM ) means precisely that F (M) is an initial object. So that if the right exact category (CX , EX ) has enough projective deflations (resp. F enough projectives), then a functor CX −→ CY is EX -coeffaceable iff F (e) is trivial for any projective deflation e (resp. F (M) is an initial object for every projective object M). ∗ 5.6.2. Universal ∂ -functors and pointed projectives. Let CZ be a category with initial objects. We call an object M of CZ pointed if there are morphisms from M to initial objects, or, equivalently, a unique morphism from an initial object to M is split.

5.6.2.1. Proposition. Let (CX , EX ) be a right exact category with initial objects ∗ and T = (Ti, di | i ≥ 0) a universal ∂ -functor from (CX , EX ) to CY . Then Ti(P ) is an initial object for any pointed projective object P and for all i ≥ 1.

55 Proof. Let F denote the functor Ti, i ≥ 0. By 3.3.2, Ti+1 ' S−(F )(P ). Let x be an initial object of CX and P a projective object of (CX , EX ) such that there exists a morphism P −→ x, then Ti+1(P ) ' S−(F )(P ) is an initial object. i id In fact, consider the conﬂation x −→P P −→P P. If there exists a morphism P −→ x,

iP then the unique arrow x −→ P is a split monomorphism. Therefore F (iP ) is a (split) monomorphism. By 2.1.1, the latter implies that Ker(F (iP )) is an initial object. Since the object P is projective, any deﬂation M −→e P is split; i.e. there exists a morphism id of deﬂations (P −→P P ) −→u (M −→e P ). This implies that the canonical morphism

S−(F )(L) −→ Ker(F (k(e)) factors through the morphism Ker(F (iP )) −→ Ker(F (k(e)) determined by the morphism of deﬂations u. Since Ker(F (iP )) = y is an initial object of the category CY , it follows that the morphism Ker(F (iP )) −→ Ker(F (k(e)) is unique (in particular, it does not depend on the choice of the section P −→u M). Therefore, the canonical morphism S−(F )(L) −→ Ker(F (iP )) = y is an isomorphism.

5.6.2.2. Corollary. Let (CX , EX ) be a right exact category with initial objects and ∗ T = (Ti, di | i ≥ 0) a universal ∂ -functor from (CX , EX ) to CY . Suppose that (CX , EX ) has enough projectives and projectives of (CX , EX ) are pointed objects. Then the functors Ti are coeﬀaceable for all i ≥ 1. Proof. The assertion follows from 5.6.2.1 and 5.6.1.

5.6.3. Proposition. Let (CX , EX ) and (CZ , EZ ) be right exact categories and ∗ f ∗ CZ −→ CX a functor having a right adjoint f∗. Suppose that f maps deflations of (M) ∗ the form N −→ f∗(M) to deflations and the adjunction arrow f f∗(M) −−−→ M is a deflation for all M (which is the case if any morphism t of CX such that f∗(t) is a split epimorphism belongs to EX ). Let (CZ , EZ ) have enough projectives, and all projectives are pointed objects. Then each projective of (CX , EX ) is a pointed object. If, in addition, f∗ maps deflations to deflations, then (CX , EX ) has enough projectives.

Proof. (a) Let M be an object of CX . Since (CZ , EZ ) has enough projectives, there t exists a deflation Pe −→ f∗(M), where Pe is a projective object. By hypothesis, the morphisms f ∗(t) (M) ∗ ∗ f (Pe) −−−→ f f∗(M) −−−→ M (1) are deflations. If the object M is projective, their composition is a split epimorphism; i.e. ζ γ it has a section M −→ f ∗(Pe). By hypothesis, there exists a morphism Pe −→ z, where z is ∗ ∗ an initial object of CZ . Since the functor f has a right adjoint, f (z) is an initial object f ∗(γ)◦ζ ∗ of the category CX , and we have a morphism M −−−→ f (z). (b) If, in addition, the functor f∗ maps deflations to deflations, then, by 5.3.1, its left adjoint f ∗ maps projectives to projectives. So that the composition of the arrows (1) is a deflation with a projective domain. 5.6.4. Note. The conditions of 5.6.3 can be replaced by the requirement that if ∗ N −→ f∗(M) is a deflation, then the corresponding morphism f (N) −→ M is a deflation.

56 This requirement follows from the conditions of 5.6.3, because the morphism f ∗(N) −→ M t ∗ corresponding to N −→ f∗(M) is the composition of f (t) and the adjunction arrow (M) ∗ f f∗(M) −−−→ M.

5.6.5. Example. Let (CX , EX ) be the category Algk of associative k-algebras endowed with the canonical (that is the finest) right exact structure. This means that class EX of deflations coincides with the class of all are strict epimorphisms of k-algebras. Let (CY , EY ) be the category of k-modules with the canonical exact structure, and f∗ the ∗ forgetful functor Algk −→ k − mod. Its left adjoint, f preserves strict epimorphisms, and the functor f∗ preserves and reflects deflations; i.e. a k-algebra morphism t is a strict epi- ∗ morphism iff f∗(t) is an epimorphism. In particular, the adjunction arrow f f∗(A) −→ A is a strict epimorphism for all A. By 5.6.3, (CX , EX ) has enough projectives and each projective has a morphism to the initial object k; that is each projective has a structure of an augmented k-algebra.

5.6.6. Proposition. Let (CX , EX ) and (CY , EY ) be right exact categories with initial ∗ objects; and let T = (Ti, di| i ≥ 0) be an ’exact’ ∂ -functor from (CX , EX ) to (CY , EY ). ∗ If the functors Ti are EX -coeﬀaceable for i ≥ 1, then T is a universal ∂ -functor. 0 0 0 ∗ Proof. Let T = (Ti , di| i ≥ 0) be another ∂ -functor from (CX , EX ) to CY and f0 a 0 j e functor morphism T0 −→ T0. Fix an object L of CX . Let N −→ M −→ L be a conﬂation such that T1(e) factors through the initial object y of CY . Then we have a commutative diagram

0 0 0 0 T1(e) d T0(j) T0(e) T 0(M) −−−→ T 0(L) −−−→ T 0(N) −−−→ T 0(M) −−−→ T 0(L) 1 1 0  0 0    f0(N) y y f0(M) y f0(L) (1) T1(e) d T0(j) T0(e) T1(M) −−−→ T1(L) −−−→ T0(N) −−−→ T0(M) −−−→ T0(L)

Since the lower row of (1) is an ’exact’ sequence and T1(e) factors through the initial object y, the sequence d T0(j) y −−−→ T1(L) −−−→ T0(N) −−−→ T0(M) f (L) 0 1 is ’exact’. Therefore, there exists a unique morphism T1(L) −−−→ T1(L) such that the diagram 0 0 d T0(j) T 0(L) −−−→ T 0(N) −−−→ T 0(M) 1 0 0    f1(L) y f0(N) y y f0(M) d T0(j) T1(L) −−−→ T0(N) −−−→ T0(M) commutes. By a standard argument, it follows from the uniqueness of f1(L) and the fact that the family of the deflations of L is filtered (since pull-backs of deflations are deflations) that the morphism f1(L) does not depend on a choice of the conflation and the 0 family f1 = (f1(L) | L ∈ ObCX ) is a functor morphism T1 −→ T1 compatible with the 0 connecting morphisms d0, d0.

57 5.6.6.1. Note. If a right exact category (CX , EX ) has enough projectives and each projective is a pointed object, then, by 5.6.2.2, for any universal ∂∗-functor T , the functors Ti are EX -coeﬀaceable for i ≥ 1.

5.6.7. Proposition. Let (CX , EX ), (CY , EY ), and (CZ , EZ ) be right exact categories. Suppose that (CX , EX ) has enough projectives and CY has kernels of all morphisms. If ∗ T = (Ti| i ≥ 0) is a universal, ’exact’ ∂ -functor from (CX , EX ) to (CY , EY ) and F a functor from (CY , EY ) to (CZ , EZ ) which respects conflations, then the composition F ◦T = ∗ (F ◦ Ti| i ≥ 0) is a universal ’exact’ ∂ -functor. Proof. Since T is a universal ∂∗-functor, it follows from 5.6.2.2 that the functors Ti are EX -coeffaceable for all i ≥ 1. Moreover, because CX has enough projectives, the coeffaceability of Ti means precisely that Ti(L) = 0 for any projective object L of (CX , EX ). ∗ Therefore, F ◦ Ti(L) = 0 for all i ≥ 1, i.e. the ∂ -functor F ◦ T is EX -effaceable. Since by hypothesis, T is an ’exact’ ∂∗-functor and F is an ’exact’ functor, their composition, F ◦ T , is an ’exact’ ∂∗-functor. By 5.6.6(a), it is universal.

5.6.8. A remark about (co)eﬀaceable functors. Let CX be a category with initial objects and B its subcategory. We say that an object M of CX is right (resp. left) orthogonal to B if for every N ∈ ObB, there are only trivial morphisms from N to M (resp. ⊥ ⊥ from M to N). We denote by B (resp. B) the full subcategory of CX generated by objects right (resp. left) orthogonal to B. Let (CX , EX ) and (CY , EY ) be right exact categories, and let y be an initial object of the category CY . The category Hom(CX ,CY ) of functors from CX to CY has an initial object, which is the constant functor with values in y. Let Rex((CX , EX ), (CY , EY )) be the full subcategory of Hom(CX ,CY ) whose objects are right ’exact’ functors. And let o Eff ((CX , EX ),CY ) denote the full subcategory of Hom(CX ,CY ) generated by coeﬀace- able functors from (CX , EX ) to CY .

5.6.8.1. Proposition. Let (CX , EX ) be a svelt right exact category with enough projectives and (CY , EY ) a right exact category with an initial object y. Suppose that CY is a category with kernels of morphisms and the unique morphism from the initial object y of o CY are monomorphisms. Then Eff ((CX , EX ),CY ) is right orthogonal to the subcategory generated by all functors CX −→ CY which map deﬂations to strict epimorphisms.

o φ Proof. Let F ∈ ObEff ((CX , EX ),CY ); and let G −→ F be a functor morphism, where G is a functor which maps deﬂations to strict epimorphisms. Since (CX , EX ) has e enough projectives, for each object L of CX , there exists a deﬂation P −→ L such that P is a projective object. Then we have a commutative diagram

φ(P ) G(P ) −−−→ F (P )     G(e) y y F (e) φ(L) G(L) −−−→ F (L) with F (P ) being an initial object of CY ; so that the composition φ(L) ◦ G(e) is a trivial morphism. By 2.1.2, all morphisms from initial objects are monomorphisms iﬀ for any

58 f morphism M −→ N the kernel morphism Ker(f) −→ M is a monomorphism. Since G(e) is a strict epimorphism and the kernel morphism Ker(φ(L)) −→ G(L) is a monomorphism, it follows from 2.3.4.4 that φ(L) is a trivial morphism.

5.6.8.2. Proposition. Let (CX , EX ) be a svelt right exact pointed category with enough projectives, and let (CY , EY ) be the category of pointed sets with the canonical F exact structure. Then a functor CX −→ CY is coeﬀaceable iﬀ it is a right orthogonal to the subcategory Rex((CX , EX ), (CY , EY )) of right ’exact’ functors from (CX , EX ) to CY .

Proof. The fact that coeffaceable functors to CY are right orthogonal to right ’exact’ functors follows from 5.6.8.1, because right ’exact’ functors map deflations to deflations, and deflations are strict epimorphisms. F Conversely, let a functor CX −→ CY be right orthogonal to all right ’exact’ functors from (CX , EX ) to CY . Notice that for any projective object P of (CX , EX ), the functor Pˇ = CX (P, −) is ’exact’, in particular it is right ’exact’. By the (dual version of) Yoneda lemma, Hom(P,Fˇ ) ' F (P ). Since, by hypothesis, Hom(P,Fˇ ) consists of the trivial morphism, F (P ) is trivial for all projective objects P of (CX , EX ). Since (CX , EX ) has enough projectives, this means precisely that F is a coeffaceable functor. The k-linear version of 5.6.8.2 is as follows.

5.6.8.3. Proposition. Let (CX , EX ) be a svelt right exact k-linear category with F enough projectives. A k-linear functor CX −→ k − mod is coeffaceable iff it is right orthogonal to the subcategory Rexk((CX , EX ), k − mod) of right ’exact’ k-linear functors from (CX , EX ) to k − mod. Proof. The argument is similar to that of 5.6.8.2. 6. Left exact categories of ’spaces’. We start with studying left exact stuctures formed by localizations of ’spaces’ represented by svelte categories. Then the obtained facts are used to define natural left exact structures on the category of ’spaces’ represented by right exact categories. The following proposition is a refinement of [R1, 1.4.1]. f q 6.1. Proposition. Let Z ←− X −→ Y be morphisms of ’spaces’ such that q (i.e. its q∗ inverse image functor CY −→ CX ) is a localization. Then q a (a) The canonical morphism Z −→e Z Y is a localization. f,q (b) If q is a continuous localization, then qe is a continuous localization. ∗ (c) If Σq∗ = {s ∈ HomCY | q (s) is invertible} is a left (resp. right) multiplicative system, then Σ has the same property. eq∗ a Y Proof. Let X denote the ’space’ Z Y . The category CX is CZ CY . Recall that f,q f ∗,g∗ Y objects of CZ CY are triples (L, M; φ), where L ∈ ObCZ ,M ∈ ObCY , and φ is an f ∗,g∗

59 isomorphism f ∗(L) −→∼ q∗(M). A morphism (L, M; φ) −→ (L0,M 0; φ0) is given by a pair β of arrows, L −→α L0 and M −→ M 0, such that the diagram

f ∗(α) f ∗(L) −−−→ f ∗(L0)     0 φ yo oy φ q∗(β) q∗(M) −−−→ q∗(M 0) commutes. The composition of morphisms is deﬁned naturally. q∗ q The (canonical) inverse image CX −→e CZ of the coprojection Z −→e X maps each (s,t) object (L, M; φ) to L and each morphism (L, M; φ) −−−→ (L0,M 0; φ0) to L −→s L0. It (s,t) follows that the class Σ consists of all morphisms (L, M; φ) −−−→ (L0,M 0; φ0) such that eq∗ s is an isomorphism, hence t ∈ Σq∗ . ∗ (a) Since q is a localication, for any L ∈ ObCZ , there exists M ∈ ObCY such that φ there is an isomorphism f ∗(L) −→ q∗(M). The map L 7−→ (L, M; φ) (– a choice for each −1 L of an object M and isomorphism φ) extends uniquely to a functor CZ −→ Σ CX which q∗ −1 e is quasi-inverse to the canonical functor Σ CX −−−→ CZ , (L, M; φ) 7−→ L. eq∗ (b) Suppose that q is a continuous localization; i.e. the localization functor q∗ has η ∗ ∗ ∗ a right adjoint, q∗. Fix adjunction arrows IdCY −→ q∗q and q q∗ −→ IdCX . Since q

q∗ is a localization, is an isomorphism. Therefore, we have a functor CZ −→e CX which ∗ ∗ maps any object L of CZ to the object (L, q∗f (L); f (L)) of the category CX and any ξ 0 ∗ morphism L −→ L to the morphism (ξ, q∗f (ξ)) of CX. ∗ The functor qe∗ is a right adjoint to the projection qe . The adjunction morphism Id −→ q q∗ assigns to each object (L, M; φ) of the category C the morphism CX e∗ e X

(id ,φ) L b ∗ ∗ (L, M; φ) −−−−−−−→ (L, q∗f (L); f (L)),

−1 φb ∗ ∗ φ ∗ where M −→ q∗f (L) denote the morphism conjugate to q (M) −→ f (L). The adjunction arrow q∗q −→ Id is the identical morphism. The latter implies that q∗ is a e e∗ CZ e localization functor. ∗ (c) Suppose that Σq∗ = {s ∈ HomCY | q (s) is invertible} is a left multiplicative (s,t) system. Let (L, M; φ) −−−→ (L0,M 0; φ0) be a morphism of Σ (that is L −→s L0 is an eq∗ (ξ,γ) 00 00 00 isomorphism) and (L, M; φ) −−−→ (L ,M ; φ ) an arbitrary morphism of CX. The claim is that there exists a commutative diagram

(ξ,γ) (L, M; φ) −−−→ (L00,M 00; φ00)     0 0 (s, t) y y (s , t ) (1) (ξ0,γ0) (L0,M 0; φ0) −−−→ (L,e Mf; φe)

60 in CX whose right vertical arrow belongs to Σ ∗ . eq t 0 In fact, since M −→ M belongs to Σq∗ and Σq∗ is a left multiplicative system, there exists a commutative diagram γ M −−−→ M 00     0 t y y t γ0 M 0 −−−→ Mf

0 00 0 ∗ 00 in CY such that t ∈ Σq∗ . Setting Le = L , s = idL00 , and φe = q (t) ◦ φ , we obtain the required commutative diagram (1). (s,t) 0 0 0 (c1) Suppose that (L, M; φ) −−−→ (L ,M ; φ ) is a morphism of Σq∗ which equal- (α,β) e 0 0 0 −−−→ 00 00 00 izes a pair of arrows (L ,M ; φ ) −−−→ (L ,M ; φ ). Then there exists a morphism (ξ,γ) (s0,t0) 00 00 00 (L ,M ; φ ) −−−→ (L,e Mf; φe) of Σ ∗ which equalizes this pair of arrows. eq In fact, since s is an isomorphism, the equality (α, β)◦(s, t) = (ξ, γ)◦(s, t) implies that α = ξ. Since Σq∗ is a left multiplicative system, the equality β ◦ t = γ ◦ t (and the fact that 0 00 t 0 0 t ∈ Σq∗ ) implies the existence of a morphism M −→ Mf in Σq∗ such that t ◦ β = t ◦ γ. 00 0 ∗ 0 00 Taking Le = L , s = idL00 , and φe = q (t ) ◦ φ , we obtain the required morphism of Σ ∗ . eq

(c’) Suppose that Σq∗ is stable under the base change. Then Σ ∗ has the same property. eq (s,t) 0 0 0 In fact, let a morphism (L ,M ; φ ) −−−→ (L, M; φ) of CX belong to Σ ∗ , and let eq (ξ,γ) 00 00 00 (L ,M ; φ ) −−−→ (L, M; φ) be an arbitrary morphism of CX. Then there exists a commutative diagram (ξ0,γ0) (L,e Mf; φe) −−−→ (L0,M 0; φ0)   0 0   (s , t ) y y (s, t) (2) (ξ,γ) (L00,M 00; φ00) −−−→ (L, M; φ) in CX whose left vertical arrow belongs to Σ ∗ . eq 0 t Since M −→ M belongs to Σq∗ and Σq∗ is a left multiplicative system, there exists a commutative diagram γ0 M −−−→ M 0 f  0   t y y t γ M 00 −−−→ Mf

0 00 0 ∗ 0 −1 00 in CY such that t ∈ Σq∗ . Setting Le = L , s = idL00 , and φe = q (t ) ◦ φ , we obtain (s0,t0) 00 00 00 0 −1 a morphism (L,e Mf; φe) −−−→ (L ,M ; φ ) which belongs to Σ ∗ . Set ξ = s ◦ ξ. The eq claim is that the pair (ξ0, γ0) is a morphism from (L,e Mf; φe) to (L0,M 0; φ0) which makes

61 the diagram (2) commute. By deﬁnition, (ξ0, γ0) being a morphism from (L,e Mf; φe) to (L0,M 0; φ0) means the commutativity of the diagram

f ∗(ξ0) f ∗(Le) −−−→ f ∗(L0)     0 φe yo oy φ q∗(γ0) q∗(Mf) −−−→ q∗(M 0) which amounts to the equalities

q∗(γ0) ◦ q∗(t0)−1 ◦ φ00 = q∗(γ0) ◦ φe = φ0 ◦ f ∗(ξ0) = φ0 ◦ f ∗(s)−1 ◦ f ∗(ξ). (3)

It follows from the equality t ◦ γ0 = γ ◦ t0 that q∗(γ0) ◦ q∗(t0)−1 = q∗(t)−1 ◦ q∗(γ). On the other hand, the fact that (s, t) is a morphism from (L0,M 0; φ0) to (L, M; φ) means that q∗(t)◦φ0 = φ◦f ∗(s), or, equivalently, φ0 ◦f ∗(s)−1 = q∗(t)−1 ◦φ. Therefore, (3) is equivalent to the equality q∗(t)−1 ◦ q∗(γ) ◦ φ00 = q∗(t)−1 ◦ φ ◦ f ∗(ξ), or q∗(γ) ◦ φ00 = φ ◦ f ∗(ξ). The latter equality expresses the fact that (ξ, γ) is a morphism from (L00,M 00; φ00) to (L, M; φ); hence (3) holds. The commutativity of the diagram (2) follows directly from the deﬁnition of the morphism (ξ0, γ0). (c”) Let Σq∗ have the property: 0 t 00−→ 0 (#) if an arrow M −→ M belongs to Σq∗ and equalizes a pair of arrows M −→M , 0 00 t 0 then there exists a morphism M −→ M in Σq∗ which equalizes this pair of arrows. (s,t) 0 0 0 Then Σq∗ has the same property; that is if (L ,M ; φ ) −−−→ (L, M; φ) is a morphism e (α,β) 00 00 00 −−−→ 0 0 0 of Σq∗ which equalizes a pair of arrows (L ,M ; φ ) −−−→ (L ,M ; φ ), then there exists e (ξ,γ) (s0,t0) 00 00 00 a morphism (L,e Mf; φe) −−−→ (L ,M ; φ ) of Σ ∗ which equalizes this pair of arrows. eq In fact, since s is an isomorphism, the equality (s, t)◦(α, β) = (s, t)◦(ξ, γ) implies that α = ξ. Since Σq∗ is a right multiplicative system, the equality t ◦ β = t ◦ γ (and the fact 0 00 t 0 0 that t ∈ Σq∗ ) implies the existence of a morphism M −→ Mf in Σq∗ such that t ◦β = t ◦γ. 00 0 ∗ 0 −1 00 Taking Le = L , s = idL00 , and φe = q (t ) ◦ φ , we obtain an object (L,e Mf; φe) and a (s0,t0) 00 00 00 morphism (L,e Mf; φe) −−−→ (L ,M ; φ ) which belongs to Σq∗ and equalizes the pair of (α,β) e 00 00 00 −−−→ 0 0 0 arrows (L ,M ; φ ) −−−→ (L ,M ; φ ). (ξ,γ)

If follows from (c’) and (c”) above that Σ ∗ is a right multiplicative system if Σq∗ is eq a right multiplicative system.

f q 6.2. Corollary. Let Z ←− X −→ Y be morphisms of ’spaces’ such that q is a q a localization, and let Z −→e Z Y be a canonical morphism. Suppose the category CY has f,q

62 ∗ ﬁnite limits (resp. ﬁnite colimits). Then qe is a left (resp. right) exact localization, if the localization q∗ is left (resp. right) exact.

∗ Proof. By 6.1(a), qe is a localization functor. q∗ Suppose that the category CY has ﬁnite limits and the localization functor CY −→ CX ∗ is left exact. Then it follows from [GZ, I.3.4] that Σq∗ = {s ∈ HomCY | q (s) is invertible} is a right multiplicative system. The latter implies, by 6.1(c), that Σq∗ is a right multi- ∗ e plicative system. Therefore, by [GZ, I.3.1], the localization functor qe is left exact. The following assertion is a reﬁnement of [R1, 1.4.2]. p q 6.3. Proposition. Let X ←− Z −→ Y be morphisms of ’spaces’ such that p∗ and q∗ are localization functors. Then the square

q Z −−−→ Y     p y y p1 q1 a X −−−→ X Y p,q is cartesian. u v Proof. Let X ←− W −→ Y be morphisms of ’spaces’ such that q1 ◦ u = p1 ◦ v. ∗ ∗ ψ ∗ ∗ s 0 In other words, there exists an isomorphism u ◦ q1 −→ v ◦ p1. Let M −→ M be ∗ any morphism of Σq∗ . Since p is a localization functor, there exists L ∈ ObCX and φ (idL,s) an isomorphism p∗(L) −→ q∗(M). We have a morphism (L, M; φ) −−−→ (L, M 0; φ0) of 0 ∗ a the category CX, where φ = q (s)φ and X denotes the ’space’ X Y represented by p,q Y ∗ ∗ the category CX = CX CY . By the deﬁnition of the canonical functors q1 and p1, p∗,q∗ ∗ ∗ ∗ ∗ ∗ we have q1 (idL, s) = idL and p1(idL, s) = s. Therefore, v (s) = v ◦ p1(idL, s) and ∗ ∗ ∗ ∗ ∗ ψ ∗ ∗ u ◦ q1 (idL, s) = u (idL) = idu∗(L). Since there is an isomorphism, u ◦ q1 −→ v ◦ p1, we have a commutative diagram

ψ(L,M;φ) u∗(L) −−−−−−−→ v∗(M)     ∗ id y y v (s) ψ(L,M 0;φ0) u∗(L) −−−−−−−→ v∗(M 0) whose horizontal arrows are isomorphisms, hence v∗(s) is an isomorphism. Thus, v∗ maps ∗ arrows of Σq∗ to isomorphisms. Since q is a localization, there exists a unique functor ∗ ev ∗ ∗ ∗ CY −→ CW such that v = ve ◦ q ; that is the morphism v is uniquely represented as the composition q ◦ w. Similarly, the morphism u is represented as the composition p ◦ ue for a unique ue. The equality q1 ◦ u = p1 ◦ v can be now rewritten as (q1 ◦ p) ◦ ue = (p1 ◦ q) ◦ ve = 63 ∗ ∗ ∗ ∗ ∗ (q1 ◦ p) ◦ v,e which means that ue ◦ (q1 ◦ p) ' ve ◦ (q1 ◦ p) . By 6.1(a), the functors q1 and ∗ ∗ ∗ ∗ ∗ p1 are localizations, hence (q1 ◦ p) = (p1 ◦ q) ' q ◦ p1 is a localization. Therefore the ∗ ∗ ∗ ∗ ∗ isomorphism ue ◦ (q1 ◦ p) ' ve ◦ (q1 ◦ p) implies (is equivalent to) that ve is isomorphic ∗ to ue , that is ue = ve. 6.4. Left exact structures on the category of ’spaces’. Let L denote the class of all localizations of ’spaces’ (i.e. morphisms whose inverse image functors are q localizations). We denote by L` (resp. Lr) the class of localizations X −→ Y of ’spaces’ ∗ such that Σq∗ = {s ∈ HomCY | q (s) is invertible} is a left (resp. right) multiplicative system. We denote by Le the intersection of L` and Lr (i.e. the class of localizations q c such that Σq∗ is a multiplicative system) and by L the class of continuous (i.e. having a c c c direct image functor) localizations of ’spaces’. Finally, we set Le = L ∩ Le; i.e. Le is the q class of continuous localizations X −→ Y such that Σq∗ is a multiplicative system.

c c 6.4.1. Proposition. Each of the classes of morphisms L, L`, Lr, Le, L , and Le are structures of a left exact category on the category |Cat|o of ’spaces’. Proof. It is immediate that each of these classes is closed under composition and contains all isomorphisms of the category |Cat|o. It follows from 6.1 that each of the classes is stable under cobase change. In other words, the arrows of each class can be regarded as cocovers of a copretopology. It remains to show that these copretopologies are subcanonical. Since L is the finest copretopology, it suffices to show that L is subcanonical. The copretopology L being subcanonical means precisely that for any localization q X −→ Y , the square q X −−−→ Y     q y y q1 q2 a Y −−−→ Y Y q,q is cartesian. But, this follows from 6.3. 6.5. Observation. Each object of the left exact category (|Cat|o, Lc) is injective. q In fact, a ’space’ X is an injective object of (|Cat|o, Lc) iff each morphism X −→ Y t is split; i.e. there is a morphism Y −→ X such that t ◦ q = idX . Since the morphism ∗ q is continuous, it has a direct image functor, q∗, which is fully faithful, because q is a ∗ localization functor. The latter means precisely that the adjunction arrow q q∗ −→ IdCX is an isomorphism. Therefore, the morphism Y −→t X whose inverse image functor coincides with q∗ satisfies the equality t ◦ q = idX . 6.6. Relative ’spaces’. The category |Cat|o has canonical initial object represented by the category with one object and one morphism, but does not have final objects (since we do not allow empty categories). In particular, the notion of the cokernel of a morphism is not defined in |Cat|o. So that we cannot apply to |Cat|o the theory of derived functors (satellites) sketched in Section 3. The category of relative ’spaces’ (i.e. ’spaces’ over a given ’space’) has both final objects and cokernels of arbitrary morphisms.

64 o Fix a ’space’ S. The category |Cat| /S has a natural ﬁnal object – (S, idS), and f cokernels of morphisms. The cokernel of a morphism (X, g) −→ (Y, h) of ’spaces’ over a a h S is the pair (Y S, eh), where Y S −→e S is the unique arrow determined by the f,g f,g commutative square f X −−−→ Y     g y y h idS S −−−→ S The canonical inverse image functor eh∗ of the morphism eh maps every object M of the ∗ ∗ ∗ ∼ ∗ a category CS to the object (h (M),M; f h (M) → g (M)) of the category CY CS f ∗,g∗ a representing the ’space’ Y S. f,g

6.6.1. Lemma. Let CX be a category and V its object. Any left exact structure IX on CX induces a left exact structure, IX /V on the category CX /V .

f 0 0 Proof. By the definition of IX /V , a morphism (L, ξ) −→ (L , ξ ) of CX /V belongs to f 0 IX /V iff the morphism L −→ L belongs to IX . We leave to the reader the verifying that IX /V is a left exact structure on CX /V . In particular, each left exact structure from the list of 7.4.1 induces a left exact structure on the category |Cat|o/S. 6.7. Left exact structures on the category of k-’spaces’. Fix a commutative associative unital ring k. Recall that k-’spaces’ are ’spaces’ represented by k-linear additive o categories. They are objects of the category |Catk| whose arrows X −→ Y are represented o by isomorphism classes of k-linear functors CY −→ CX . The category |Catk| is pointed: f o its zero object is represented by the zero category. Every morphism X −→ Y of |Catk| c ∗ has a canonical cokernel Y −→ Cok(f), where CCok(f) is the subcategory Ker(f ) of CY (– the full subcategory generated by all objects L such that f ∗(L) = 0) and c∗ is the ∗ inclusion functor Ker(f ) −→ CY . c c o Each of the left exact structures L, L`, Lr, Le, L , and Le on the category |Cat| of o ’spaces’ (see 6.4) induces a left exact structure on the category |Catk| of k-spaces. Thus, c c o we have left exact structures L(k), L`(k), Lr(k), Le(k), L (k), and Le(k) on |Catk| . 6.8. Left exact structures on the category of right (or left) exact ’spaces’. A right exact ’space’ is a pair (X, EX ), where X is a ’space’ and EX is a right exact structure on the category CX . We denote by Espr the category whose objects are right exact ’spaces’ (X, EX ) and morphisms from (X, EX ) to (Y, EY ) are given by morphisms f X −→ Y of ’spaces’ whose inverse image functor, f ∗, is ’exact’; i.e. f ∗ maps deflations to deflations and preserves pull-backs of deflations. Dually, a left exact ’space’ is a pair (Y, IY ), where (CY , IY ) is a left exact category. We denote by Esp` the category whose objects are left exact ’spaces’ (Y, IY ) and morphisms

65 (Y, IY ) −→ (Z, IZ ) are given by morphisms Y −→ Z whose inverse image functors are ’coexact’, which means that they preserve arbitrary push-forwards of inﬂations.

6.8.1. Note. The categories Espr and Esp` are naturally isomorphic to each other: o op the isomorphism is given by the dualization functor (X, EX ) 7−→ (X , EX ). Therefore every assertion about the category Espr of right exact ’spaces’ translates into an assertion about the category Esp` of left exact ’spaces’ and vice versa.

6.8.2. Proposition. The category Espr has fibered coproducts. f g Proof. Let (X, EX ) ←− (Z, EZ ) −→ (Y, EY ) be morphisms of Espr; and let X denote a Y the ’space’ X Y , i.e. CX = CX CY . Let EX denote the class of all morphisms f,g f ∗,g∗ (ξ,γ) 0 0 0 ξ 0 γ 0 (L, M; φ) −−−→ (L ,M ; φ ) of CX such that L −→ L belongs to EX and M −→ M is an arrow of EY . The claim is that EX is a right exact structure on CX and (X, EX) is a a coproduct (X, EX ) (Y, EY ) of right exact ’spaces’. f,g It is immediate that EX contains all isomorphisms and is closed under composition. Let (ξ,γ) (α,β) 0 0 0 00 00 00 0 0 0 (L, M; φ) −−−→ (L ,M ; φ ) be a morphism of EX, and let (L ,M ; φ ) −−−→ (L ,M ; φ ) ∗ ∗ be an arbitrary morphism of CX. Since the inverse image functors f and g preserve corresponding deflations and their pull-backs and ξ, γ are deflations, the isomorphisms φ, φ0, φ Y Y and φ00 induce an isomorphism f ∗(Le) −→e g∗(Mf), where Le = L L00 and Mf = M M 00. ξ,α γ,β It is easy to see that the square

(α0,β0) (L,e Mf; φe) −−−→ (L, M; φ)     (ξ,e γe) y y (ξ, γ) (1) (α,β) (L00,M 00; φ00) −−−→ (L0,M 0; φ0) is cartesian, ξe ∈ EX , and γe ∈ EY . Therefore, (ξ,e γe) ∈ EX. If (α, β) = (ξ, γ), then the square (1) is cocartesian, because the squares

ξ0 γ0 L −−−→ L M −−−→ M e  f      ξe y y ξ and γe y y γ ξ γ L −−−→ L0 M −−−→ M 0 are (both cartesian and) cocartesian. Altogether shows that the arrows of EX are covers of a subcanonical pretopology; i.e. EX is a structure of a right exact category on CX.

6.8.3. Canonical left exact structures on the category Espr. Let Les denote q ∗ the class of all morphisms (X, EX ) −→ (Y, EY ) of right exact ’spaces’ such that q is a ∗ localization functor and each arrow of EX is isomorphic to an arrow q (e) for some e ∈ EY .

66 If Σq∗ is a left or right multiplicative system, then this condition means that EX is ∗ the smallest right exact structure containing q (EY ).

6.8.3.1. Proposition. The class Les is a left exact structure on the category Espr of right exact ’spaces’.

Proof. The class Les contains, obviously, all isomorphisms, and it is easy to see that it is closed under composition. It remains to show that Les is stable under cobase change and its arrows are cocovers of a subcanonical copretopology. q f Let (X, EX ) −→ (Y, EY ) be a morphism of Les and (X, EX ) −→ (Z, EZ ) an arbitrary q a morphism. The claim is that the canonical morphism Z −→e Z Y belongs to Les. f,q Consider the corresponding cartesian (in pseudo-categorical sense) square of right exact categories: p∗ (CX, EX) −−−→ (CY , EY )   ∗   ∗ qe y y q (2) f ∗ (CZ , EZ ) −−−→ (CX , EX )

a Y ∗ where X = Z Y ; that is CX = CZ CY . Recall that the functor qe maps each object f,q f ∗,q∗ (L, M; φ) of the category CX to the object L of CZ and each morphism (ξ, γ) to ξ. By ∗ ∗ 6.1(a), qe is a localization functor (because q is a localization functor). e 0 ∗ q Let L −→ L be an arrow of EZ . Then f (e) is a morphism of EX . Since X −→ Y is t 0 a morphism of Les, there exists a morphism M −→ M of EY and a commutative diagram

f ∗(e) f ∗(L) −−−→ f ∗(L0)     0 ψ yo oy ψ q∗(t) q∗(M) −−−→ q∗(M 0) whose vertical arrows are isomorphisms. By the deﬁnition of the right exact category 0 0 0 (CX, EX), this means that (e, t) is a morphism (L, M; ψ) −−−→ (L ,M ; ψ ) of CX which ∗ ∗ belongs to EX. The localization functor qe maps it to e. Thus, EZ = qe (EX), hence qe ∈ EX. This shows that Les is stable under cobase change. q It remains to verify that for every morphism (X, EX ) −→ (Y, EY ) of Les the square

∗ p1 (CY, EY) −−−→ (CY , EY )   ∗   ∗ p2 y y q (3) q∗ (CY , EY ) −−−→ (CX , EX ) Y is cocartesian. Here CY = CY CY . q∗,q∗

67 Consider a quasi-commutative diagram

∗ p1 (CY, EY) −−−→ (CY , EY )   ∗   ∗ p2 y y v (4) u∗ (CY , EY ) −−−→ (CW , EW ) of ’exact’ functors. Since, by 6.3, that the square

∗ p1 CY −−−→ CY   ∗   ∗ p2 y y q q∗ CY −−−→ CX

w∗ is cocartesian, there exists a unique up to isomorphism functor CX −→ CW such that ∗ ∗ ∗ ∗ ∗ v ' w q ' u . The claim is that w is an ’exact’ functor from (CX , EX ) to (CW , EW ). ∗ ∗ Since q ∈ Les, every morphism of EX is isomorphic to a morphism of q (EY ) and v maps ∗ ∗ ∗ ∗ ∗ EY to EW . Therefore w maps EX to EW . The fact that q and v ' w q are ’exact’ functors implies that the functor w∗ is ’exact’.

c 6.8.3.2. Corollary. Each of the classes of morphisms of ’spaces’ L`, Lr, Le, L , c and Le (cf. 6.4, 6.4.1) induces a structure of a left exact category on the category Espr of right exact ’spaces’.

es Proof. The class L` induces the class L` of morphisms of the category Espr formed q q by all arrows (X, EX ) −→ (Y, EY ) from Les such that the morphism of ’spaces’ X −→ Y es es c e,c belongs to L`. Similarly, we deﬁne the classes L` , Lr , Les, and Les . es 6.8.3.3. The left exact structure Lsq. For a right exact ’space’ (X, EX ), let Sq(X, EX ) denote the class of all cartesian squares in the category CX some of the arrows of which (at least two) belong to EX . es q The class Lsq consists of all morphisms (X, EX ) −→ (Y, EY ) of right exact ’spaces’ such that its inverse image functor, q∗, is equivalent to a localization functor and each ∗ square of Sq(X, EX ) is isomorphic to some square of q (Sq(Y, EY )). es 6.8.3.4. Proposition. The class Lsq is a left exact structure on the category Espr es of right exact ’spaces’ which is coarser than Les and ﬁner than Lr . Proof. The argument is left to the reader.

6.9. Relative right exact ’spaces’. The category Espr of right exact ’spaces’ has initial objects and no final object. Final objects appear if we fix a right exact ’space’ S = (S, ES ) and consider the category Espr/S instead of Espr. The category Espr/S has a natural final object and cokernels of all morphisms. It also inherits left exact structures from Espr, in particular those defined above (see 6.8.3.2). Therefore, our theory of derived functors (satellites) can be applied to functors from Espr/S.

68 6.10. The category of right exact k-’spaces’. For a commutative unital ring k, r we denote by Espk the category whose objects are right exact ’spaces’ (X, EX ) such that CX is a k-linear additive category and morphisms are morphisms of right exact ’spaces’ whose inverse image functors are k-linear. es es es c e,c Each of the left exact structures Les, L` , Lr , Lr , Les, and Les induces a left exact r structure on the category Espk of right exact k-’spaces’. We denote them by respectively es es es c e,c Les(k), L` (k), Lr (k), Lr (k), Les(k), and Les (k). 6.11. The path ’space’ of a right exact ’space’. Fix a right exact svelte category (CX , EX ). Let CX be the quotient of the category CPa(X) of paths of the category CX by the relations s ◦ef = f ◦ t, where f N −−−→e M e    t y cart y s f N −−−→ L runs through cartesian squares in CX whose vertical arrows belong to EX . In particular, ObCX = ObCX . We denote by EX the image in CX of all paths of morphisms of EX and by Pa(X, EX ) the pair (X, EX).

6.11.1. Proposition. Let (CX , EX ) be a svelte right exact category and (X, EX) = Pa(X, EX ) (see above). (a) The class of morphisms EX is a right exact structure on the category CX. ∗ εX (b) The canonical functor CPa(X) −−−→ CX (identical on objects and mapping paths ∗ pX of arrows to their composition) factors uniquely through a functor CX −−−→ CX which p X es is an inverse image functor of a morphism (X, EX ) −−−→ (X, EX) that belongs to Lsq. (c) The right exact ’space’ (X, EX) is an injective object of a left exact category es (Espr, Lsq). Proof. (a) It follows (from the fact that the composition of cartesian squares is a cartesian square) that EX is a right exact structure on CX. ∗ εX (b) The functor CPa(X) −−−→ CX is (equivalent to) a localization functor which p∗ X ∗ factors uniquely through CX −−−→ CX . Therefore, pX is (equivalent to) a localization ∗ functor. It follows from deﬁnitions that pX maps cartesian squares with deﬂations among their arrows to cartesian squares of the same type. Moreover, all cartesian squares with pX this property are obtained this way. Therefore, the morphism (X, EX ) −→ (X, EX) belongs es to the class Lsq. q es f (c) Let (Z, EZ ) −→ (Y, EY ) be a morphism of Lr and (Z, EZ ) −→ (X, EX) an ar- γ bitrary morphism.The claim is that there exists a morphism (Y, EY ) −→ (X, EX) of right exact ’spaces’ such that f = γ ◦ q. q∗ f∗ (c1) Let (CY , EY ) −→ (CZ , EZ ) and (CX, EX) −→ (CZ , EZ ) be inverse image functors

69 of respectively q and f. Consider the standard cartesian square

∗ eq (CY, EY) −−−→ (CX, EX)   ∗   ∗ f1 y y f q∗ (CY , EY ) −−−→ (CZ , EZ )

q of right exact categories. By 6.8.3.4, the morphism (X, EX) −→e (Y, EY) represented by ∗ ∗ es eq the functor eq belongs to Lsq. Moreover, the functor CY −→ CX is surjective. es In fact, q being a morphism of Lsq implies that every square of Sq(Z, EZ ) is isomor- ∗ phic to q (e) for some e ∈ Sq(Y, EY ). In particular, every morphism of CZ is isomorphic ξ to the image of some arrow of CY . Thus, for any morphism M −→ L of CX, there is a commutative diagram f∗(ξ) f∗(M) −−−→ f∗(L)     φ yo oy ψ q∗(γ) q∗(V ) −−−→ q∗(W ) whose vertical arrows are isomorphisms. In other words, the pair (ξ, γ) is a morphism ∗ (M,V ; φ) −→ (L, W ; ψ) of the category CY, and ξ = eq (ξ, γ). ∗ (c2) The functor eq maps Sq(Y, EY) onto Sq(X, EX). (c3) Given a class of arrows S in HomCY , we denote by RS the class of all pairs of arrows M ⇒ L which are equalized by an arrow N −→ M from S. If S contains all identical morphisms, closed under composition, and ﬁltered, i.e. every pair of arrows L −→ M ←− N of S can be completed to a commutative square

L −−−→ L e    y y N −−−→ M whose arrows belong to S, then RS is an equivalence relation. ∗ (c4) In particular, RΣq∗ is an equivalence relation, because Σq is a right multiplicative system. For every L ∈ ObCY , we denote by EY,L the class of all deﬂations of L. For each

L ∈ ObCY , we choose representatives of equivalence classes with respect to RΣq∗ of arrows to L. 6.12. Complements. E 6.12.1. The left exact structure Lr . Fix a right exact category (CX , EX ). We say that a class Σ of deﬂations is EX -saturated if it is the intersection of a saturated system of arrows of CX and EX .

6.12.1.1. Lemma. Let Σ be an EX -saturated class of deﬂations. Then Σ is a right multiplicative system iﬀ it is stable under base change.

70 Proof. Let Σ be an EX -saturated system of deﬂations. In particular, it contains all isomorphisms of CX and is closed under compositions. If Σ is stable under base change, it is a right multiplicative system. Conversely, if Σ is a right multiplicative system, then, by [GZ, I.3.1], the localization ∗ q −1 functor CX −→ Σ CX is right exact. In particular, it maps all cartesian squares of CX −1 to cartesian squares of Σ CX . Since EX is stable under base change, every diagram s f M −→ L −→ N with s ∈ EX can be completed to a cartesian square

f N −−−→e M e    t y y s (1) f N −−−→ L

∗ and t ∈ EX . If s ∈ Σ, then the localization q maps (1) to a cartesian square whose right vertical arrow, q∗(s), is an isomorphism. Therefore its left vertical arrow is an isomorphism. Since Σ is EX -saturated, this implies that t ∈ Σ. s We denote by S Mr(X, EX ) the preorder (under the inclusion) of all EX -saturated right multiplicative systems Σ of EX having the following property: (#) If the right horizontal arrows in the commutative diagram

p0 1 −−−→ ee Mf −−−→ M −−−→ L p0  2      0 et y y s y s p1 −−−→ e M ×L M −−−→ M −−−→ L p2 are deflations, the pairs of arrows are kernel pairs of these deflations and two left vertical arrows belong to Σ, then the remaining vertical arrow belongs to Σ. q 6.12.1.2. Proposition. (a) For any morphism (Y, EY ) −→ (X, EX ) of the category T ∗ Espr of right exact ’spaces’, the intersection Σq∗ EX = {t ∈ EX | q (t) is invertible} s belongs to S Mr(X, EX ). ∗ s q −1 (b) For any Σ ∈ S Mr(X, EX ), the localization functor CX −→ Σ CX = CX is an st q st inverse image functor of a morphism (X, EX ) −→ (X, EX ) of Espr. As usual, EX denote the finest right exact structure on CX.

Proof. (a) By definition of morphisms of the category Espr, its inverse image functor maps pull-backs of deflations to pull-backs of deflations. Therefore the intersection T ∗ Σq∗ EX = {t ∈ EX | q (t) is invertible } is (by definition) saturated and stable under base change. The property (#) follows from the ’exactness’ of the localization functor q∗. s (b) Let Σ ∈ S Mr(X, EX ). Since Σ is a right multiplicative system, the localization ∗ q −1 functor CX −→ Σ CX = CX is left exact. In particular, it maps all cartesian squares to cartesian squares. It remains to show that it maps deflations to strict epimorphisms.

71 p e −→1 Let M −→ L be a morphism of EX and M ×L M −→ M its kernel pair. Let p2 0 p1 ∗ ξ ∗ ∗ −→ q (M) −→ q (N) be a morphism which equalizes the pair q M ×L M −→ M . Since Σ p2 is a right multiplicative system, the morphism ξ0 is the composition q∗(ξ)q∗(s)−1 for some ξ morphisms M ←−s M −→ N, where s ∈ Σ. Thus we have a diagram

u1 u2 M −−−→ M × M ←−−− M 1 L  2    t1y cart p1 yy p2 cart y t2 s s M −−−→ M ←−−− M whose both squares are cartesian, all arrows are deﬂations, and all horizontal arrows belong to Σ. Therefore, there exists a cartesian square 0 v1 Mf −−−→ M2     v2 y cart y u2 u1 M1 −−−→ M ×L M whose all arrows belong to Σ. Altogether leads to a commutative diagram

p0 1 −−−→ ee Mf −−−→ M −−−→ L p0  2      0 et y y s y s p1 −−−→ e M ×L M −−−→ M −−−→ L p2 whose rows are exact diagrams and two (left) vertical arrows belong to Σ. Therefore, the remaining vertical arrow belongs to Σ. The localization functor q∗ maps the compositions 0 0 ξ◦p1 and ξ◦p2 to the same arrow. This means precisely that there exists a morphism λ ∈ Σ 0 0 such that ξ ◦p1 ◦λ = ξ ◦p2 ◦λ (cf. [GZ, I.2.2]). Since all morphisms of Σ are epimorphisms, p0 −→1 the latter equality implies that the morphism ξ equalizes the pair Mf−→M. Therefore, p0 2 ee it factors uniquely through the morphism M −→ L; i.e. ξ = ξe◦ e. The pair of arrows 0 ξ L ←−s L −→e N determines a unique morphism q∗(L) −→ q∗(N) whose composition with q∗(e) equals to ξ0.

E q We denote by Lr the class of all morphisms (X, EX ) −→ (Y, EY ) whose inverse image s functor is equivalent to the localization functor at a system which belongs to S Mr(X, EX ). E 6.12.1.3. Proposition. The class of morphisms Lr is a left exact structure on the category Espr of right exact ’spaces’. E Proof. The class of morphisms Lr contains all isomorphisms and is closed under compositions and cobase change.

72 7. K-theory of right exact ’spaces’.

7.1. The functor K0.

7.1.1. The group Z0|CX |. For a svelte category CX , we denote by |CX | the set of isomorphism classes of objects of CX , by Z|CX | the free abelian group generated by |CX |, and by Z0(CX ) the subgroup of Z|CX | generated by diﬀerences [M] − [N] for all arrows M −→ N of the category CX . Here [M] denotes the isomorphism class of an object M.

7.1.2. Proposition. (a) The maps X 7−→ Z|CX | and X 7−→ Z0(CX ) extend naturally to presheaves of Z-modules on the category of ’spaces’ |Cat|o (i.e. to functors from (|Cat|o)op to Z − mod). (b) If the category CX has an initial (resp. final) object x, then Z0(CX ) is the subgroup of Z|CX | generated by differences [M] − [x], where [M] runs through the set |CX | of isomorphism classes of objects of CX . Proof. The argument is left to the reader. op 7.1.3. Remarks. (a) Evidently, there are natural isomorphisms Z|CX | ' Z|CX | and op Z0(CX ) ' Z0(CX ). (b) Let Z0(CX ) be regarded as a groupoid with one object, •. Then the map which assigns to every object of CX the object • and to any morphism M −→ N of CX the difference [M] − [N] is a functor from CX to the groupoid Z0(CX ).

7.1.4. The group K0 of a right exact ’space’. Let (X, EX ) be a right exact ’space’. We denote by K0(X, EX ) the quotient of the group Z0|CX | by the subgroup generated by the expressions [M 0] − [M] + [L] − [N] for all cartesian squares

f M 0 −−−→e M   0   e y cart y e f L0 −−−→ L whose vertical arrows are deﬂations. We call K0(X, EX ) the group K0 of the right exact ’space’ (X, EX ).

7.1.4.1. Example: the group K0 of a ’space’. Any ’space’ X is identiﬁed with the trivial right exact ’space’ (X, Iso(CX )). We set K0(X) = K0(X, Iso(CX )). That is K0(X) coincides with the group Z0(CX ).

7.1.5. Proposition. (a) The map (X, EX ) 7−→ K0(X, EX ) extends to a contravariant functor, K0, from the category Espr of right exact ’spaces’ (cf. 6.8) to the category Z − mod of abelian groups. f (b) Let (X, EX ) −→ (Y, EY ) be a morphism of Espr having the following property: 0 0 (†) if M and L are non-isomorphic objects of CX which can be connected by non- oriented sequence of arrows (i.e. they belong to one connected component of the associated groupoid), then there exist objects M and L of CY which have the same property and such that f ∗(M) ' M 0, f ∗(L) ' L0.

73 Then K0(f) K0(Y, EY ) −−−→ K0(X, EX ) is a group epimorphism. In particular, the functor K0 maps ’exact’ localizations to epimorphisms.

f ∗ Proof. (a) Let (X, EX ) and (Y, EY ) be right exact ’spaces’ and (CY , EY ) −→ (CX , EX ) an ’exact’ functor. Then f ∗ induces a morphism

K0(f) K0(Y, EY ) −−−→ K0(X, EX ) uniquely determined by the commutativity of the diagram

∗ Z0(f ) Z0(CY ) −−−→ Z0(CX )     pY y y pX (1) K0(f) K0(Y, EY ) −−−→ K0(X, EX )

∗ of Z-modules. Here Z0(f ) denotes the morphism of abelian groups induced by the functor ∗ f . The vertical arrows, pY and pX , are natural epimorphisms. f (b) Suppose that (X, EX ) −→ (Y, EY ) is a morphism of Espr having the property (†). (f ∗) Z0 ∗ Then Z0(CY ) −−−→ Z0(CX ) is a group epimorphism. Thus, K0(f) ◦ pY = pX ◦ Z0(f ) is an epimorphism, which implies that K0(f) is an epimorphism. f 7.1.5.1. Corollary. Let (X, EX ) −→ (Y, EY ) be a morphism of Espr whose inverse ∗ image functor, f , induces a surjective map |CY | −→ |CX | of isomorphism classes of objects. If the groupoid associated with the category CY is connected, then K0(f) is a surjective map. In particular, K0(f) is surjective if the category CY has initial or ﬁnal objects. Proof. The assertion follows from 7.1.5(b).

q ∗ 7.1.5.2. Corollary. For any ’exact’ localization (X, EX ) −→ (Y, EY ) (i.e. q is equivalent to a localization functor), the map K0(q) is an epimorphism. ∗ Proof. If q is equivalent to a localization functor, then each object of CX is isomorphic ∗ ∗ ∗ to an object of q (CY ) and any morphism q (M) −→ q (L) is the composition of the form ∗ −1 ∗ ∗ −1 ∗ q (sn) ◦ q (fn) ◦ · · · ◦ q (s1) ◦ q (f1) for some chain of arrows

f1 s1 f1 fn sn M −→ Mf1 ←− M1 −→ ... −→ Mfn ←− Mn = L.

In particular, the condition (†) of 7.1.5(b) holds.

7.1.6. Proposition. Let (X, EX ) be a right exact ’space’ such that the category CX has initial objects. Then the group K0(X, EX ) is the quotient of the free abelian group Z|CX | generated by the isomorphism classes of objects of CX by the subgroup generated by

74 [M] − [L] − [N] for all conﬂations N −→ M −→ L and the isomorphism class of initial objects of CX . Proof. (a) The expressions [M] − [L] − [N], where N −→k M −→e L runs through conﬂations of (CX , EX ), are among the relations because each of them corresponds to a cartesian square N −−−→ x     k y cart y e M −−−→ L where x is an initial object of CX . (b) On the other hand, let

e M −−−→e L f e 0   (1) f y cart y f e M −−−→ L be a cartesian square whose horizontal arrows are deflations. Therefore we have a commutative diagram k e N −−−→e M −−−→e L  f e  0   id y f y cart y f k e N −−−→ M −−−→ L whose rows are conflations. The rows give relations [Mf] − [Le] − [N] and [M] − [L] − [N]. Their difference, [Mf]−[M]+[L]−[Le], is the relation corresponding to the cartesian square (1). Hence the assertion. w ∗ w 7.1.7. The categories Espr and Espr . Let Espr denote the category whose objects are right exact ’spaces’ (X, EX ) such that CX has initial objects; and morphisms f (X, EX ) −→ (Y, EY ) are given by morphisms of ’spaces’ X −→ Y whose inverse image functors preserve conflations. In particular, they map initial objects to initial objects. ∗ We denote by Espr the subcategory of Espr whose objects are right exact ’spaces’ (CX , EX ) such that the category CX has initial objects and morphisms are defined by the requirement that their inverse image functor maps initial objects to initial objects. ∗ w It follows that Espr is a subcategory of the category Espr . The k-linear versions of these categories coincide.

7.1.8. Proposition. (a) The map (X, EX ) 7−→ K0(X, EX ) extends to a contravari- w w ant functor, K0 , from the category Espr to the category Z − mod of abelian groups. f w ∗ (b) Let (X, EX ) −→ (Y, EY ) be a morphism of Espr such that f induces a surjective map |CY | −→ |CX | of the isomorphism classes of objects. Then

Kw(f) w 0 w K0 (Y, EY ) −−−→ K0 (X, EX )

75 is a group epimorphism. In particular, the functor K0 maps ’exact’ localizations to epimorphisms.

Proof. The assertions follow from 7.1.6.

7.2. The relative functors K0 and their derived functors. Fix a right exact K op 0 ’space’ Y = (Y, EY ). The functor (Espr) −−−→ Z − mod induces a functor

KY op 0 (Espr/Y) −−−→ Z − mod defined by K (ξ) Y Y ξ 0 K0 (X , ξ) = K0 (X , X → Y) = Cok(K0(Y) −−−→ K0(X )) and acting correspondingly on morphisms. Y The main advantage of the functor K0 is that its domain, the category Espr/Y has a final object, cokernels of morphisms, and natural left exact structures induced by left exact structures on Espr. Fix a left exact structure I on Espr (say, one of those defined in 6.8.3.2) and denote by IY the left exact structure on Espr/Y induced by I. Notice that, since the category Z − mod is complete (and cocomplete), there is a well defined satellite op endofunctor of Hom((Espr/Y) , Z − mod),F 7−→ SIY F. So that for every functor F op ∗ from (Espr/Y) to Z − mod, there is a unique up to isomorphism universal ∂ -functor (Si F, d | i ≥ 0). IY i ∗ Y,I Y,I In particular, there is a universal contravariant ∂ -functor K• = (Ki , di | i ≥ 0) from the right exact category (Espr/Y, IY ) of right exact ’spaces’ over Y to the category − mod of abelian groups; that is KY,I = Si KY,I for all i ≥ 0. Z i IY 0 Y,I We call the groups Ki (X , ξ) universal K-groups of the right exact ’space’ (X , ξ) over Y with respect to the left exact structure I.

∗ Y,I 7.3. ’Exactness’ properties. In general, the ∂ -functor K• is not ’exact’. The purpose of this section is to ﬁnd some natural left exact structures I on the category ∗ Espr/Y of right exact ’spaces’ over Y and its subcategory Espr /Y (cf. 7.1.7) for which the ∗ Y,I ∂ -functor K• is ’exact’.

q 0 0 7.3.1. Proposition. Let (X, ξ) −→ (X , ξ ) be a morphism of the category Espr/Y q 0 such that X −→ X belongs to Les (cf. 6.8.3) and has the following property: s ∗ (#) if M −→ L is a morphism of CX0 such that q (s) is invertible, then the element K0(cq ) 0 00 0 [M] − [L] of the group K0(X ) belongs to the image of the map K0(X ) −−−→ K0(X ), c q where (X0, ξ0) −→q (X00, ξ00) is the cokernel of the morphism (X, ξ) −→ (X0, ξ0). Suppose, in addition, that one of the following two conditions holds: (i) the category CX0 has an initial object; f s ∗ (ii) for any pair of arrows N −→ L ←− M, of the category CX0 such that q (s) is

76 invertible, there exists a commutative square

f N −−−→e M e    t y y s f N −−−→ L such that q∗(t) is invertible. q c Then for every conﬂation (X, ξ) −→ (X0, ξ0) −→q (X00, ξ00) of the left exact category (Espr/Y, IY ) the sequence

Y Y K (cq) K (q) Y 00 00 0 Y 0 0 0 Y K0 (X , ξ ) −−−→ K0 (X , ξ ) −−−→ K0 (X, ξ) −−−→ 0 of morphisms of abelian groups is exact.

KY (q) Y 0 0 0 Y Proof. (a) The map K0 (X , ξ ) −−−→ K0 (X, ξ) is surjective, because, by 7.1.5.2, the K (q) 0 0 map K0(X , EX0 ) −−−→ K0(X, EX ) is surjective. c q (b) Fix a cokernel (X0, ξ0) −→q (X00, ξ00) of (X, ξ) −→ (X0, ξ0) and its inverse image ∗ cq functor CX00 −→ CX0 . Notice that the condition (#) is equivalent to the condition ∗ ∗ (#’) If M and L are objects of CX0 such that q (M) ' q (L), then [M] − [L] belongs to Im(K0(cq)). Obviously, (#’) implies (#). On the other hand, since q∗ is a localization functor, the existence of an isomorphism between q∗(M) and q∗(L) is equivalent to the existence of a diagram M ←− M1 −→ M2 ←− ... ←− Mn −→ L ∗ whose arrows belong to Σq∗ = {s ∈ HomCX0 | q (s) is invertible}, which shows that (#) implies the condition (#’). (b1) The condition (#’) is equivalent to the condition ∗ Z0(q ) (#”) The kernel of the morphism Z0(CX0 ) −−−→ Z0(CX ) is contained in the subgroup ∗ Im(Z0(cq )) + Ker(pX0 ) of the abelian group Z0CX0 ). 0 Here pX0 is the canonical epimorphism Z0(CX0 ) −→ K0(X ). In fact, the condition (#’) follows from (#”): it suﬃces to apply (#”) to the elements ∗ of Ker(Z0(q )) of the form [M] − [L]. X An element z = λ[M][M] of the abelian group Z0(CX0 ) can be written as

[M]∈|CX0 | X X ∗ λ[M][M]. It follows that the element z belongs to Ker(Z0(q )) ∗ N∈|CX | {[M]|[q (M)]=N} X iﬀ λ[M] = 0 for each N ∈ |CX |. It follows from the condition (#’) that {[M]|[q∗(M)]=N} X ∗ λ[M][M] belongs to the subgroup Im(Z0(cq )) + Ker(pX0 ) of the group {[M]|[q∗(M)]=N}

77 X ∗ Z0(CX0 ) whenever λ[M] = 0. Therefore, each element of Ker(Z0(q )) be- {[M]|[q∗(M)]=N} ∗ longs to the subgroup Im(Z0(cq )) + Ker(pX0 ). (c) Consider the commutative diagram

0 0 0    y y y K(cq) K(q) Ker(pX00 ) −−−→ Ker(pX0 ) −−−→ Ker(pX )    00  0   jX y jX y y jX (c∗) ∗ Z0 q Z0(q ) (1) Z0(CX00 ) −−−→ Z0(CX0 ) −−−→ Z0(CX )    00  0   pX y pX y y pX K0(cq ) K (q) 00 0 0 K0(X ) −−−→ K0(X ) −−−→ K0(X)       y y y 0 0 0 with exact columns. K(q) (c1) The map Ker(pX0 ) −−−→ Ker(pX ) is surjective. 0 q In fact, by hypothesis, the localization X −→ X belongs to Les; that is each morphism ∗ of EX is isomorphic to an arrow of q (EX0 ). (i) Suppose that the category CX0 has initial objects. Since q is a morphism of the ∗ category Espr, its inverse image functor, q , maps initial objects to initial objects (in ∗ particular, the category CX has initial objects, which is also a consequence of q being a localization functor) and conflations to conflations. Therefore, any conflation of the ∗ right exact category (CX , EX ) is isomorphic to q (N −→ M −→ L) for some conflation N −→ M −→ L of the right exact category (CX0 , EX0 ). So that the subgroup Ker(pX ) is generated by the elements K(q)([M] − [N] − [L]), where N −→ M −→ L runs through K(q) conflations of (CX0 , EX0 ), whence the surjectivity of Ker(pX0 ) −−−→ Ker(pX ). (ii) Suppose now that the condition (ii) holds. The claim is that, in this case, every cartesian square in CX whose vertical arrows are deflations is isomorphic to to the image of a cartesian square in CX0 with the same property. ∗ ∗ Since each arrow of EX is isomorphic to an arrow of q (EX ) and q is a localization functor, every cartesian square in CX is isomorphic to the cartesian square of the form

fe q∗(Ne) −−−→ q∗(M)     ∗ (2) u y cart y q (e) f q∗(N) −−−→ q∗(L) ∗ where e ∈ EX0 . The functor q being a localization implies that the morphism f is the composition ∗ ∗ −1 ∗ ∗ −1 ∗ ∗ −1 q (fn)q (sn) . . . q (f2)q (s2) q (f1)q (s1) .

78 By the condition (2), there exists a commutatative square

fe1 Ne1 −−−→ L     t1y y s1 f1 N1 −−−→ L1

∗ ∗ ∗ −1 ∗ −1 such that q (t1) is an isomorphism. Therefore, q (f1)q (s1) = q (t1) fe1, which implies ∗ ∗ −1 ∗ ∗ −1 ∗ that f = q (fn)q (sn) . . . q (f2)q (t1s2) q (fe1). Continuing this process, we obtain f 0 morphisms N ←−t N −→ L such that q∗(t) is an isomorphism and f = q∗(t)−1q∗(f 0). Since the morphism e in the diagram (2) is a deﬂation, there is a cartesian square

f 00 M −−−→ M     e y y e f 00 N −−−→ L

Since the functor q∗ is ’exact’ it maps this cartesian square to a cartesian square which is (thanks to the iniversal property of cartesian squares) isomorphic to the square (2). 0 This shows that the subgroup Ker(pX ) is generated by the elements K(q)([M ]−[M]+ [L] − [N]) for all cartesian squares

f M 0 −−−→e M   0   e y cart y e f L0 −−−→ L

K(q) whose vertical arrows are deﬂations. Hence the surjectivity of Ker(pX0 ) −−−→ Ker(pX ). (c2) There is the inclusion Ker(K0(q)) ⊆ Im(K0(cq)). 0 0 0 Indeed, let z ∈ K0(X ), and let z be an element of Z0(CX0 ) such that pX0 (z ) = z. The ∗ 0 element z belongs to Ker(K0(q)) iﬀ the element Z0(q )(z ) belongs to Ker(pX ). Thanks 00 to the surjectivity of K(q) (argued in (c1)), there is an element z in Ker(pX0 ) such that 00 ∗ 0 ∗ 0 00 K(q)(z ) = Z0(q )(z ). Therefore, Z0(q )(z − z ) = 0. By the property (#”), which is equivalent to the property (#) of the proposition ∗ Z0(q ) (see (b1) above), the kernel of the morphism Z0(CX0 ) −−−→ Z0(CX ) is contained in the ∗ subgroup Im(Z0(cq )) + Ker(pX0 ) of the abelian group Z0(CX0 ). So that

0 00 ∗ ∗ z ∈ z + Im(Z0(cq )) + Ker(pX0 ) ⊆ Im(Z0(cq )) + Ker(pX0 ).

0 ∗ Therefore, z = pX0 (z ) ∈ pX0 (Im(Z0(cq ))) ⊆ Im(K0(cq)).

79 (c3) It follows from the commutative diagram

K (q) 00 0 e K0(X ) −−−→ K0(Y )     K0(cq) y y K0(ξ) K (q) λ(q) 0 0 Y K0(X ) −−−→ K0(X) −−−→ K (X, ξ) = Cok(K0(ξ)) that the inclusion Ker(K0(q)) ⊆ Im(K0(cq)) implies the exactness of the sequence

K (c ) λ(q)K (q) 00 0 q 0 0 Y K0(X ) −−−→ K0(X ) −−−→ K (X, ξ) −−−→ 0 which, in turn, implies the exactness of the sequence

Y Y K (cq) K (q) Y 00 00 0 Y 0 0 0 Y K0 (X , ξ ) −−−→ K0 (X , ξ ) −−−→ K0 (X, ξ) −−−→ 0 as claimed. Y q 0 0 7.3.2. Proposition. The class Les of all morphisms (X, ξ) −→ (X , ξ ) of Espr/Y q 0 such that X −→ X belongs to Les and satisﬁes the condition (#) of 7.3.1, is a left exact structure on the category Espr/Y. Y Proof. It is clear that Les contains all isomorphisms. We need to show that it is stable under cobase change and closed under compositions. q 0 0 Y f (i) Let (X, ξ) −→ (X , ξ ) be a morphism of Les and (X, ξ) −→ (Z, ζ) an arbitrary morphism of Espr/Y. We have a quasi-commutative diagram

∗ λp C 00 −−−→ C Z Y ∗  ∗ cpy cart y ζ ∗ ∗ cp p CZ00 −−−→ CZ0 −−−→ CZ 00  0   ∗ ∗  ∗ f yo f y cart y f (3) ∗ ∗ cq q C 00 −−−→ C 0 −−−→ C X xX xX ∗  ∗ cq cart  ξ ∗ λq CX00 −−−→ CY whose right squares are cartesian and (therefore) the left vertical arrow is a category (s,t) 0 0 0 equivalence. A morphism (M,L; φ) −−−→ (M ,L ; φ ) belongs to Σp∗ iﬀ t is an isomorphism s 0 and, therefore, M −→ M belongs to Σq∗ . Since we are looking at isomorphism classes of objects, we can and will assume that L = L0 and t is the identity morphism. 0 By the condition (#) of 7.3.1, the fact that s ∈ Σq∗ implies that [M] − [M ] ∈ c∗ ∗ ∗ p f Z0(cq). The composition of CZ00 −→ CZ0 with the isomorphism CX00 −→ CZ00 maps

80 ∗ ψ ∗ ∗ ∗ ∼ ∗ ∗ each object (M, K; q (M) → ξ (K)) of CX00 to the object (M, ζ (K); q (M) → f ζ (K)) 0 ∗ of the category CZ0 . Therefore, the inclusion [M] − [M ] ∈ Z0(cq) implies the inclusion 0 0 ∗ [(M,L; φ)] − [(M ,L; φ )] ∈ Z0(cp). p q Y (ii) Let (X, ξ) −→ (X, ξX) and (X, ξX) −→ (Z, ξZ ) be morphisms of Les. The claim qp Y is that their composition, (X, ξ) −→ (Z, ξZ ), belongs to Les too; that is the localization qp X −→ Z (which belongs to Les) satisﬁes the property (#) of 7.3.1. We have the quasi- commutative diagram

CZ000 −−−→ CY ∗   c  cart  ξX00 γq y y c∗ ∗ ∗ γq γq λp C 000 −−−→ C 00 −−−→ C 00 −−−→ C Z Z X Y  ∗   ∗  ∗ oy cqpy cart y cp cart y ζ (4) ∗ ∗ ∗ cq q p C 00 −−−→ C −−−→ C −−−→ C X xZ xX X ∗   cq  cart  ξX CX00 −−−→ CY with cartesian squares as indicated. To this diagram, there corresponds the commutative diagram id Z0(CX00 ) −−−→ Z0(CX00 )   ∗  ∗  Z0(β ) y Z0(cq) y ∗ ∗ 0(c ) ((qp) ) Z qp Z0 (5) Z0(CZ00 ) −−−→ Z0(CZ ) −−−→ Z0(CX )    ∗  ∗   Z0(γq ) y Z0(q ) y y id (c∗ ) ∗ Z0 p Z0(p ) Z0(CX00 ) −−−→ Z0(CX) −−−→ Z0(CX ) ∗ ∗ of abelian groups with surjective vertical arrows Z0(γq ) and Z0(q ). Here the functor c∗ ∗ γq β is the composition of the equivalence CX00 −→ CZ000 and the functor CZ000 −→ CZ00 ∗ ∗ ∗ (see the diagram (4)). Let z ∈ Ker(Z0(cqp)), or, equivalently, Z0(q )(z) ∈ Ker(Z0(p )). p Y Since (X, ξ) −→ (X, ξX) belongs to Les, it follows from the condition (#) of 7.3.1 that ∗ ∗ ∗ Ker(Z0(p )) ⊆ Im(Z0(cp)) (see the argument of 7.3.1). Since Z0(γq ) is an epimorphism, ∗ ∗ there exists an element b ∈ Z0(CZ00 ) such that z − Z0(cqp)(b) ∈ Ker(Z0(q ). Since q ∗ ∗ satisﬁes the condition (#) of 7.3.1, Ker(Z0(q )) ⊆ Im(Z0(cq)), and it follows from the ∗ ∗ upper square of (5) that Im(Z0(cq)) ⊆ Im(Z0(cqp)). All together implies that z is an ∗ element of Im(Z0(cqp)); i.e. the condition (#) holds. This shows that the composition q◦p Y (X, ξ) −−−→ (Z, ξZ ) belongs to the class Les. Y (iii) The argument above proves that Les is the class of covers of a copretopology. This copretopology is subcanonical (i.e. it is a left exact structure on the category Espr/Y), because the copretopology Les on Espr is subcanonical.

81 Y q 0 0 Y 7.3.2.1. Proposition. The class Les,r of all morphisms (X, ξ) −→ (X , ξ ) og Les q∗ such that the functor CX0 −→ CX satisﬁes the condition (ii) of 7.3.1, is a left exact structure on the category Espr/Y. Proof. The assertion follows from 7.3.2 and the part (c) of the argument of 6.1.

7.3.3. Proposition. Let Y = (Y, EY ) be a right exact ’space’, and let I be a left exact Y structure on the category Espr/Y which is coarser than Les,r (cf. 7.3.2). Then the universal ∗ Y Y ∂ -functor K• = (Ki , di | i ≥ 0) from the left exact category (Espr/Y, IY ) to the category q cq Z−mod of abelian groups is ’exact’; i.e. for any conﬂation (X, ξ) −→ (X0, ξ0) −→ (X00, ξ00), the associated long sequence

Y Y Y K (q) d K (cq) K (q) 1 Y 0 Y 00 00 0 Y 0 0 0 Y ... −−−→ K1 (X, ξ) −−−→ K0 (X , ξ ) −−−→ K0 (X , ξ ) −−−→ K0 (X, ξ) −−−→ 0 is exact. Y Proof. Since the left exact structure IY is coarser than Les, it satisﬁes the condition q c (#) of 7.3.1. Therefore, by 7.3.1, for any conﬂation (X, ξ) −→ (X0, ξ0) −→q (X00, ξ00) of the ∗ left exact category (Espr /Y, IY ), the sequence

Y Y K (cq) K (q) Y 00 00 0 Y 0 0 0 Y K0 (X , ξ ) −−−→ K0 (X , ξ ) −−−→ K0 (X, ξ) −−−→ 0

∗ Y Y of Z-modules is exact. Therefore, by 3.6.1, the universal ∂ -functor K• = (Ki , di| i ≥ 0) ∗ from (Espr /Y, IY ) to Z − mod is ’exact’. It is convenient to have the following generalization of the previous assertion.

7.3.4. Proposition. Let Y = (Y, EY ) be a right exact ’space’, (CS, IS) be a left exact category, and F a functor CS −→ Espr/Y which maps conflations of (CS, IS) to Y conflations of the left exact category (Espr/Y, Les,r). Then there exists a (unique up to ∗ S,F S,F isomorphism) universal ∂ -functor K• = (Ki , di | i ≥ 0) from the right exact category op S,F (CS, IS) to Z − mod whose zero component, K0 , is the composition of the functor Fop op op Y CS −−−→ Espr/Y and the functor K0 . ∗ S,F The ∂ -functor K• is ’exact’. ∗ S,F Proof. The existence of the ∂ -functor K• follows, by 3.3.2, from the completeness (– existence of limits of small diagrams) of the category Z − mod of abelian groups. The S,F main thrust of the proposition is in the ’exactness’ of K• . By hypothesis, the functor F maps conflations to conflations. Therefore, it follows 0 00 from 7.3.1 that for any conflation X −→ X −→ X of the left exact category (CS, IS), S,F 00 S,F 0 S,F the sequence of abelian groups K0 (X ) −→ K0 (X ) −→ K0 (X) −→ 0 is exact. By ∗ Y 3.6.1, this implies the ’exactness’ of the ∂ -functor K• . o 7.4. The ’absolute’ case. Let |Cat∗| denote the subcategory of the category |Cat|o of ’spaces’ whose objects are ’spaces’ represented by ’spaces’ with initial objects

82 and morphisms are those morphisms of ’spaces’ whose inverse image functor maps initial o objects to initial objects. The category |Cat∗| is pointed: it has a canonical zero (that is both initial and final) object, x, which is represented by the category with one (identical) morphism. Thus, the final objects of the category |Cat|o of all ’spaces’ are zero objects of o the subcategory |Cat∗| . f o cf Each morphism X −→ Y of the category |Cat∗| has a cokernel, Y −→ C(f), where ∗ ∗ the category CC(f) representing the ’space’ C(f) is the kernel Ker(f ) of the functor f . ∗ By definition, Ker(f ) is the full subcategory of the category CY generated by all objects ∗ ∗ of CY which the functor f maps to initial objects. The inverse image functor cf of the ∗ canonical morphism cf is the natural embedding Ker(f ) −→ CY . ∗ The category Espr formed by right exact ’spaces’ with initial objects and morphisms whose inverse image functor is ’exact’ and maps initial objects to initial objects (cf. 7.1.7), is pointed and the forgetfull functor

J∗ ∗ o Espr −−−→ |Cat∗| , (X, EX ) 7−→ X,

o J∗ ∗ is a left adjoint to the canonical full embedding |Cat∗| −→ Espr which assigns to every ∗ ’space’ X the right exact category (X, Iso(CX )). Both functors, J and J∗, map zero objects to zero objects. ∗ ∗ Let x be a zero object of the category Espr . Then Espr /x is naturally isomorphic to ∗ x Espr and K0 = K0. ∗ ∗ 7.4.1. The left exact structure Les. We denote by Les the canonical left exact x structure Les; it does not depend on the choice of the zero object x. It follows from ∗ q the deﬁnitions above that Les consists of all morphisms (X, EX ) −→ (Y, EY ) having the following properties: q∗ (a) CY −→ CX is a localization functor (which is ’exact’ and maps initial objects to ∗ initial objects), and every arrow of EX is isomorphic to an arrow of q (EY ). s 0 ∗ 0 (b) If M −→ M is an arrow of CY such that q (s) is an isomorphism, then [M]−[M ] ∗ is an element of KerK0(q )|.

7.4.2. Proposition. Let (CS, IS) be a left exact category, and let F be a functor ∗ CS −→ Espr which maps conflations of (CS, IS) to conflations of the left exact category ∗ ∗ ∗ op (Espr , Les). Let G be a functor from (Espr ) to a category CZ with limits of ’small’ filtered systems and initial objects. Then ∗ S,F S,F op (a) There exists a universal ∂ -functor G• = (Gi , edi | i ≥ 0) from (CS, IS) to Fop S,F op ∗ op CZ whose zero component, K0 , is the composition of the functor CS −−−→ (Espr ) and the functor G. ∗ (b) If (CZ , EZ ) is a right exact category and the functor G is left ’exact’, then the ∂ - S,F ∗ ∗ ∗ functor G• is ’exact’. In particular, the ∂ -functor G• = (Gi, di | i ≥ 0) from (Espr , Les) to (CZ , EZ ) is ’exact’. Proof. The assertion is a special case of 7.3.4.

83 F ∗ 7.4.2.1. Corollary. Let (CS, IS) be a left exact category, and CS −→ Espr a functor which maps conﬂations of (CS, IS) to conﬂations of the left exact category ∗ ∗ ∗ S,F (Espr , Les). Then there exists a (unique up to isomorphism) universal ∂ -functor K• = S,F op S,F (Ki , edi | i ≥ 0) from (CS, IS) to Z − mod whose zero component, K0 , is the com- Fop op ∗ op position of the functor CS −−−→ (Espr ) and the functor K0. ∗ S,F ∗ The ∂ -functor K• is ’exact’. In particular, the ∂ -functor K• = (Ki, di| i ≥ 0) ∗ ∗ from (Espr , Les) to Z − mod is ’exact’.

7.4.3. The class of morphisms Les~ . We denote by Les~ the class of all morphisms q ∗ ∗ (X, EX ) −→ (Y, EY ) of Les such that Cok(q) is a zero object, or, equivalently, Ker(q ) is q a trivial category. It follows that Les~ consists of all morphisms (X, EX ) −→ (Y, EY ) such that q∗ (a) CY −→ CX is an ’exact’ localization functor with a trivial kernel, and every arrow ∗ of EX is isomorphic to an arrow of q (EY ). ∗ s 0 0 (b) If q (M −→ M ) is an isomorphism, then [M] = [M ] in K0(Y ). ∗ 7.4.4. Proposition. The class Les~ is a left exact structure on the category Espr of right exact ’spaces’ with initial objects. Proof. The assertion is a special (dual) case of 5.3.7.1.

∗ 7.4.5. Proposition. Let (CS, IS) be a left exact category, F a functor CS −→ Espr ∗ ∗ which maps conﬂations of (CS, IS) to conﬂations of the left exact category (Espr , Les), S,F S,F ∗ op and K• = (Ki , edi | i ≥ 0) a universal ∂ -functor from (CS, IS) to Z − mod whose Fop S,F op ∗ op zero component, K0 , is the composition of CS −−−→ (Espr ) and K0 (cf. 7.4.2.1). q If X −→ Y is a morphism of IS with trivial cokernel, then the morphisms

KS,F S,F i S,F Ki (Y) −−−→ Ki (X ) are isomorphisms for all i ≥ 0.

~ Proof. Let IS denote the class of all morphisms of IS having a trivial cokernel. By ~ (the dual version of) 3.3.7.1, the class IS is a left exact structure on the category CS. F ∗ Since the functor CS −→ Espr maps conflations to conflations, it maps final objects ∗ ~ of the category CS to zero objects of Espr . In particular, F maps morphisms of IS ∗ to morphisms of Les~ . By 7.4.2.1, the ∂ -functor is ’exact’, so that for any conflation q c X −→ X 0 −→q X 00, the sequence

S,F S,F K (cq) K (q) S,F 00 0 S,F 0 0 S,F K0 (X ) −−−→ K0 (X ) −−−→ K0 (X ) −−−→ 0

~ S,F 00 00 is exact. If q ∈ IS, then K0 (X ) = K0(F(X )) = 0. So that in this case the morphism KS,F(q) S,F 0 0 S,F K0 (X ) −−−→ K0 (X ) is an isomorphism. The assertion follows now from 5.3.7.2.

84 q 0 7.4.6. Corollary. For every morphism (X, EX ) −→ (X , EX0 ) of Les~ the corresponding map K (q) 0 i Ki(X , EX0 ) −−−→ Ki(X, EX ) is an isomorphism for all i ≥ 0.

a 7.5. Universal K-theory of abelian categories. Let Espk denote the category whose objects are ’spaces’ X represented by k-linear abelian categories and morphisms f X −→ Y are represented by k-linear exact functors. There is a natural functor F a ∗ Espk −−−→ Espr (1)

a which assigns to each object X of the category Espk the right exact (actually, exact) ’space’ st st (X, EX ), where EX is the canonical (i.e. the ﬁnest) right exact structure on the category f st f st CX , and maps each morphism X −→ Y to the morphism (X, EX ) −→ (Y, EY ) of right a exact ’spaces’. One can see that the functor F maps the zero object of the category Espk ∗ (represented by the zero category) to a zero object of the category Espr .

7.5.1. Proposition. Let CX and CY be k-linear abelian categories endowed with q∗ the canonical exact structure. Any exact localization functor CY −→ CX satisﬁes the conditions (a) and (b) of 7.4.1.

Proof. In fact, each morphism q∗(M) −→eh q∗(N) is of the form q∗(h)q∗(s)−1 for some morphisms M 0 −→h N and M 0 −→s M such that q∗(s) is invertible. The morphism h is a (unique) composition j ◦ e, where j is a monomorphism and e is an epimorphism. Since the functor q∗ is exact, q∗(j) is a monomorphism and q∗(e) is an epimorphism. Therefore, eh is an epimorphism iﬀ q∗(j) is an isomorphism. This shows that the condition (a) holds. Let M −→s M 0 be a morphism and

0 −→ Ker(s) −→ M −→s M 0 −→ Cok(s) −→ 0 the associated with s exact sequence. Representing s as the composition, j◦e, of a monomorphism j and an epimorphism e, we obtain two short exact sequences,

0 −→ Ker(s) −→ M −→e N −→ 0 and 0 −→ N −→j M 0 −→ Cok(s) −→ 0, hence [M] = [Ker(s)]+[N] and [M 0] = [N]+[Cok(s)], or [M 0] = [M]+[Ker(s)]−[Cok(s)] ∗ ∗ in K0(Y ). It follows from the exactness of the functor q that the morphism q (s) is an isomorphism iﬀ Ker(s) and Cok(s) are objects of the category Ker(q∗). Therefore, in this 0 ∗ case, it follows that [M ] = [M] modulo Z|Ker(q )| in K0(Y ). q 7.5.2. Proposition. (a) The class La of all morphisms X −→ Y of the category ∗ a q a Espk such that CY −→ CX is a localization functor is a left exact structure on Espk.

85 F a ∗ (b) The functor Espk −−−→ Espr is an ’exact’ functor from the left exact category a a ∗ ∗ a −1 ∗ (Espk, L ) to the left exact category (Espr , Les). Moreover, L = F (Les), that is the left a ∗ exact structure L is induced by the left exact structure Les via the functor F. a Proof. (i) The category Espk has push-forwards. f g a In fact, for any pair of arrows X ←− Z −→ Y of Espk, consider the cartesian (in the pseudo-categorical sense) square

∗ p1 C −−−→ C X X ∗   ∗ p2 y y f (2) g∗ CY −−−→ CZ of inverse image functors. One can see from the description of the category CX and functors ∗ ∗ ∗ ∗ p1 and p2 that CX is a k-linear abelian category and the functors p1 and p2 are k-linear and exact, because f ∗ and g∗ have this property. Since the square (2) is cartesian in pseudo- categorical sense, it is cartesian in the category formed by k-linear abelian categories and k-linear exact functors. Therefore, the corresponding commutative square of ’spaces’

f Z −−−→ X     g y y p1 p2 Y −−−→ X is cocartesian. a (ii) It follows from the construction of push-forwards in Espk that the functor F −1 ∗ a preserves cocartesian squares. It is clear that F (Les) ⊆ L . On the other hand, by a ∗ 7.5.1, the functor F maps morphisms of L (– exact localizations) to morphisms of Les. a −1 ∗ Therefore L = F (Les). Since the functor F maps cocartesian squares to cocartesian a a squares, it follows that L is a left exact structure on the category Espk. a 7.5.3. The Grothendieck functor. The composition K0 of the functor

Fop a op ∗ op (Espk) −−−→ (Espr )

K∗ ∗ op 0 a and the functor (Espr ) −−−→ Z − mod assigns to each object X of the category Espk ∗ st the abelian group K0 (X, EX ) which coincides with the Grothendieck group of the abelian a category CX . We call K0 the Grothendieck functor. ∗ a a a 7.5.4. Proposition. There exists a universal ∂ -functor K• = (Ki , di | i ≥ 0) from a a op the right exact category (Espk, L ) to the category Z − mod whose zero component is the ∗ a Grothendieck functor K0. The universal ∂ -functor K• is ’exact’; that is for any exact q localization X −→ X0, the canonical long sequence

Ka(q) da(q) Ka(c ) Ka(q) 1 a 0 a 00 0 q a 0 0 a ... −−−→ K1 (X) −−−→ K0 (X ) −−−→ K0 (X ) −−−→ K0 (X) −−−→ 0 (3)

86 is exact. F a ∗ Proof. By 7.5.2(b), the functor Espk −−−→ Espr is an ’exact’ functor from the left a a ∗ ∗ exact category (Espk, L ) to the left exact category (Espr , Les) which maps the zero object a of the category Espk (– the ’space’ represented by the zero category) to a zero object of ∗ the category Espr . Therefore, F maps conﬂations to conﬂations. The assertion follows now from 7.4.2.1 applied to the functor F.

∗ a 7.5.5. The universal ∂ -functor K• and the Quillen’s K-theory. For a ’space’ Q X represented by a svelte k-linear abelian category CX , we denote by Ki (X) the i-th Q Quillen’s K-group of the category CX . For each i ≥ 0, the map X 7−→ Ki (X) extends naturally to a functor KQ a op i (Espk) −−−→ Z − mod It follows from the Quillen’s localization theorem [Q, 5.5] that for any exact localization dQ(q) q 0 Q i Q 00 X −→ X and each i ≥ 0, there exists a connecting morphism Ki+1(X) −−−→ K0 (X ), ∗ where CX00 = Ker(q ), such that the sequence

KQ(q) dQ(q) KQ(c ) KQ(q) 1 Q 0 Q 00 0 q Q 0 0 Q ... −−−→ K1 (X) −−−→ K0 (X ) −−−→ K0 (X ) −−−→ K0 (X) −−−→ 0 (4) is exact. It follows (from the proof of the Quillen’s localization theorem) that the con- Q necting morphisms di (q), i ≥ 0, depend functorially on the localization morphism q. In Q Q Q ∗ other words, K• = (Ki , di | i ≥ 0) is an ’exact’ ∂ -functor from the left exact category a a op (Espk, L ) to the category Z − mod of abelian groups. ∗ Q Naturally, we call the ∂ -functor K• the Quillen’s K-functor. a a a ∗ a a op Since K• = (Ki , di | i ≥ 0) is a universal ∂ -functor from (Espk, L ) to Z − mod, Q a ∗ the identical isomorphism K0 −→ K0 extends uniquely to a ∂ -functor morphism

ϕQ Q • a K• −−−→ K• . (5)

7.5.6. Remark. There is a canonical functorial morphism of the universal determi- det Q nant group K1 (X) (introduced by Bass [Ba, p. 389]) to the Quillen’s K1 (X). If X is aﬃne, i.e. CX is the category of modules over a ring, this morphism is an isomorphism. It is known [Ger, 5.2] that if CX is the category of coherent sheaves on the complete non- det Q singular curve of genus 1 over C, then K1 (X) −→ K1 (X) is not a monomorphism. In det a det Q particular, the composition K1 (X) −→ K1 (X) of the morphism K1 (X) −→ K1 (X) ϕQ(X) Q 1 a and the canonical morphism K1 (X) −−−→ K1 (X) is not a monomorphism. r 7.6. Universal K-theory of k-linear right exact categories. Let Espk denote the category whose objects are right exact ’spaces’ (X, EX ), where the ’space’ X is represented by a k-linear svelte additive category and morphisms (X, EX ) −→ (Y, EY ) are given by f morphisms of ’spaces’ X −→ Y whose inverse image functors are k-linear ’exact’ functors.

87 By 1.4, the ’exactness’ of a morphism f means precisely that its inverse image functor, f ∗, maps conflations to conflations. There is a natural functor F r r ∗ Espk −−−→ Espr (1) r which maps objects and morphisms of the category Espk to the corresponding objects and ∗ morphisms of the category Espr . F r r ∗ 7.6.1. Proposition. The functor Espk −−−→ Espr preserves cocartesian squares r ∗ and maps the zero object of the category Espk to the zero object of the category Espr . Proof. The argument is similar to that of 7.5.2(b). Details are left to the reader. r −1 ∗ 7.6.2. Corollary. The class of morphisms Lk = Fr (Les) is a left exact structure r r r on the category Espk and Fr is an ’exact’ functor from the left exact category (Espk, Lk) ∗ ∗ to the left exact category (Espr , Les). −1 Proof. Since the functor Fr preserves cocartesian squares, the preimage Fr (τ) of ∗ r any copretopology τ on Espr is a copretopology on the category Espk. In particular, r −1 ∗ r Lk = Fr (Les) is the class of cocovers of a copretopology. The copretopology Lk is r r subcanonical, i.e. Lk is a left exact structure on the category Espk. q It follows from the definition of the functor Fr that a morphism (X, EX ) −→ (Y, EY ) r r of Espk is a localization iff Fr(q) is a localization. In particular, Lk consists of localiza- r q tions. The copretopology Lk is subcanonical iff for any morphism (X, EX ) −→ (Y, EY ) the cocartesian square q (X, EX ) −−−→ (Y, EY )     q y y p1 (2) p2 (Y, EY ) −−−→ (X, EX) is cartesian, or, equivalently, the diagram

p1 q −−−→ (X, EX ) −−−→ (Y, EY ) −−−→ (X, EX) (3) p2 is exact. The claim is that, indeed, the diagram (3) is exact. f In fact, let (Z, EZ ) −→ (Y, EY ) be a morphism which equalizes the pair of arrows p −→1 (Y, EY ) −→ (X, EX). Since the functor Fr transforms (2) into a cartesian square, there p2 h exists a unique morphism Fr(Z, EZ ) −→ Fr(X, EX ) such that Fr(q) ◦ h = Fr(f). It follows ∗ that the inverse image h of h is a k-linear functor CX −→ CZ . Therefore h is the image of (a uniquely determined) morphism (Z, EZ ) −→ (X, EX ), hence the morphism f factors q uniquely through (X, EX ) −→ (Y, EY ). r r 7.6.3. The functor K0. We denote by K0 the composition of the functor

Fop r op r ∗ op (Espk) −−−→ (Espr )

88 K∗ ∗ op 0 and the functor (Espr ) −−−→ Z − mod. ∗ r r r 7.6.4. Proposition. There exists a universal ∂ -functor K• = (Ki , di | i ≥ 0) from r r op the right exact category (Espk, L ) to the category Z − mod whose zero component is r ∗ r the functor K0. The universal ∂ -functor K• is ’exact’; that is for any exact localization q 0 r (X, EX ) −→ (X , EX0 ) which belongs to L , the canonical long sequence

r r r K (q) K (cq) d (q) r 1 r 0 1 r 00 1 K (X.EX ) ←−−− K (X , EX0 ) ←−−− K (X , EX00 ) ←−−− ... 1  1 1 r  d0(q) y (4) r r K (cq) K (q) r 00 0 r 0 0 r K0(X , EX00 ) −−−→ K0(X , EX0 ) −−−→ K0(X, EX ) −−−→ 0 is exact. F r ∗ Proof. The functor Espk −−−→ Espr is an ’exact’ functor from the left exact category r r ∗ ∗ (Espk, L ) to the left exact category (Espr , Les) which maps the zero object of the category r ∗ Espk (– the ’space’ represented by the zero category) to a zero object of the category Espr . Therefore, F maps conﬂations to conﬂations. It remains to apply 7.4.2.1.

7.6.5. Proposition. Let (CX , EX ) be a right exact k-linear additive category, ∗ γX (CXe , EXe ) the associated exact k-linear category, and (CX , EX ) −−−→ (CXe , EXe ) the canonical fully faithful ’exact’ universal functor (see 2.6.1) regarded as an inverse image γX functor of a morphism (Xe, EXe ) −−−→ (X, EX ). K0(γX )

The map K0(X, EX ) −−−→ K0(Xe, EXe ) is a group epimorphism.

Proof. The assertion follows from the description of the exact category (CXe , EXe ) (see the argument of 2.6.1). Details are left to the reader. 7.6.6. The category of exact k-’spaces’ and Grothendieck-Quillen functor. e r Let Espk denote the full subcategory of the category Espk whose objects are pairs (X, EX ) such that (CX , EX ) is an exact k-linear category. J∗ e r It follows from 2.6.1 that the inclusion functor, Espk −−−→ Espk has a right adjoint, J∗ which assigns to each right exact k-space (X, EX ) the associated exact k-space (Xe, EXe ). ∗ r The adjunction arrow J J −→ Id r assigns to each object (X, E ) of Esp the mor- ∗ Espk X k γ X ∗ phism (X , E ) −−−→ (X, E ) (see 7.6.5). The adjunction morphism Id e −→ J J is e Xe X Espk ∗ the identity morphism. J r ∗ e Thus, Espk −−−→ Espk is a localization functor. According to 7.6.5, the functor Kr r op 0 (Espk) −−−→ Z − mod factors through the localization functor

Jop r op ∗ e op (Espk) −−−→ (Espk) .

r e op e That is the functor K0 is isomorphic to the composition K0 ◦ J∗ , where K0 denote the r e op r ∗op restriction of K0 to the subcategory (Espk) , i.e. the composition K0 ◦ J .

89 e For each exact k-space (X, EX ), the group K0(X, EX ) coincides with the Grothendieck group K0 of the exact category (CX , EX ) as it was deﬁned by Quillen [Q]. e r r 7.6.7. Proposition. The restriction L of the left exact structure L on Espk to the e e subcategory Espk is a left exact structure on Espk. J∗ e r Proof. The inclusion functor Espk −−−→ Espk preserves all colimits; in particular, −1 it preserves cocartesian squares. The latter implies that Le = J∗ (Lr) is a left exact r structure on Espk. ∗ e e e e e op In particular, we have a universal ∂ -functor K• = (Ki , di | i ≥ 0) from (Espk, L ) to Z − mod. e 7.6.8. Remarks on K-theory of k-linear exact categories. The category Espk of exact k-spaces has an automorphism D which assigns to each ’space’ (X, EX ) the dual o op ’space’ (X, EX ) represented by the opposite exact category (CX , EX ) . e 7.6.8.1. Proposition. Let F be a contravariant functor from the category Espk of exact k-’spaces’ to a category CZ with ﬁltered limits. If for each ’space’ (X, EX ), there o is an isomorphism F (X, EX ) ' F ((X, EX ) ) functorial in (X, EX ), then the universal ∗ Le ∂ -functor S• F is isomorphic to its composition with the duality automorphism D of the e category Espk. Proof. The argument is left to the reader. 7.6.8.2. Corollary. There is a natural isomorphism of universal ∂∗-functors

e e K• ' K• ◦ D.

o Proof. In fact, K0(X, EX ) is naturally isomorphic to K0((X, EX ) ), because the (iden- ∼ op op tical) isomorphism ObCX −→ Ob(CX ) implies a canonical isomorphism Z|CX | ' Z|CX |, relations defining K0 correspond to conflations, and the dualization functor D induces an isomorphism between the corresponding categories of conflations. 7.7. Digression: non-additive exact categories.

7.7.1. Deﬁnition. We call a right exact category (CX , EX ) (and the corresponding right exact ’space’ (X, EX )) an exact category (resp. an exact ’space’), if the Yoneda embedding induces an equivalence of (CX , EX ) with a fully exact subcategory of the right st exact category (CXE , E ) of sheaves on (CX , EX ). Let Espe denote the full subcategory of the category Espr of right exact ’spaces’ generated by exact ’spaces’.

J∗ 7.7.2. Proposition. The inclusion functor Espe −→ Espr has a right adjoint.

Proof. This right adjoint, J∗, assigns to each right exact ’space’ (X, EX ) the ’space’

(Xe, EXe ), where CXe is the smallest fully exact subcategory of the right exact category of sheaves on (CX , EX ) endowed with the induced right exact structure.

90 7.8. Reduction by resolution.

7.8.1. Proposition. Let (CX , EX ) be a right exact category with initial objects and CY its fully exact subcategory such that 0 00 0 (a) If M −→ M −→ M is a conﬂation with M ∈ ObCY , then M ∈ ObCY . 00 00 (b) For any M ∈ ObCX , there exists a deﬂation M −→ M with M ∈ ObCY . Then the morphism K•(Y, EY ) −→ K•(X, EX ) is an isomorphism.

q∗ Proof. Let (CZ , EZ ) −→ (CY , EY ) be an ’exact’ functor. Then we have a commutative diagram j∗ C −−−→e C Z Z ∗   ∗ (1) q y y p CY −−−→ CX

j∗ where CY −→ CX is the inclusion functor and the category CZ is defined as follows. 00 00 Its objects are triples (M ,M; t), where M ∈ ObCX ,M ∈ ObCZ , and t is a deflation j∗q∗(M) −→ M 00). Morphisms are defined naturally. The functor ej∗ maps each object L ∗ ∗ ∗ of CZ to (j q (L),L; id) and acts correspondingly on morphisms. The functor p is the 00 00 projection (M ,M; t) 7−→ M . The right exact structures on CX and CZ induce a right exact structure EZ on CZ such that all functors of the diagram (1) become ’exact’. It follows from the condition (b) that if the functor q∗ is essentially surjective on ∗ p ∗ objects, then the functor CZ −→ CX has the same property. If q is an inverse image ? ∗ functor of a morphism of Les, then same holds for p . K0(j) It follows from the conditions (a) and (b) that the map K0(Y, EY ) −−−→ K0(X, EX ) is surjective. In fact, let N −→ M −→ L be conflation in CX . Thanks to the condition (b), it can be inserted into a commutative diagram

M −−−→ M 0 −−−→ L     y cart y M −−−→ L where all arrows are deﬂations, the square is cartesian, and L, M are objects of the subcategory CY . Therefore, we obtain a commutative diagram

N −−−→ M −−−→ L e f e    y y y N −−−→ M −−−→ L (2)       y y y N −−−→ M −−−→ L whose rows and columns are conﬂations. Therefore,

[M] − [L] − [N] = ([M] − [L] − [M]) − ([Mf] − [Le] − [Ne]).

91 It follows from the condition (a) (applied to the columns of the diagram (2)) that two upper rows of (2) are conflations in (CY , EY ). Since the kernel of the map K0(j) consists of combinations (with coefficients in Z) of the expressions [M] − [L] − [N], where N −→ M −→ L runs through conflations of (CX , EX ), it follows that these combinations are equal to zero.

7.8.1.1. Note. The ﬁrst part of the argument of 7.8.1 shows that if CY is a fully exact subcategory of a right exact category (CX , EX ) satisfying the condition (b) and F0 is op a functor from Espr to a category with ﬁltered limits such that F0(Y, EY ) −→ F0(X, EX ) n n is an isomorphism, then S−F0(Y, EY ) −→ S−F0(X, EX ) is an isomorphism for all n ≥ 0. The condition (a) was used only in the proof that K0(Y, EY ) −→ K0(X, EX ) is an isomorphism.

7.8.2. Proposition. Let (CX , EX ) and (CZ , EZ ) be right exact categories with initial ∗ objects and T = (Ti, di | i ≥ 0) an ’exact’ ∂ -functor from (CX , EX ) to (CZ , EZ ). Let CY be the full subcategory of CX generated by T -acyclic objects (that is objects V such that Ti(V ) is an initial object of CZ for i ≥ 1). Assume that for every M ∈ ObCX , there is a deﬂation P −→ M with P ∈ ObCY , and that Tn(M) is an initial object of CZ for n suﬃciently large. Then the natural map K•(Y, EY ) −→ K•(X, EX ) is an isomorphism.

Proof. Let CYn denote the full subcategory of the category CX generated by all objects M such that Ti(M) is an initial object of CZ for i ≥ n.

(i) All the subcategories CYn are fully exact. Indeed, if N −→ M −→ L is a conﬂation in (CX , EX ) such that N and L are objects ∗ of the subcategory CYn , then, thanks to the ’exactness’ of the ∂ -functor T , we have an exact sequence

... −→ Tm+1(L) −→ Tm(N) −→ Tm(M) −→ Tm(L) −→ ...

If m ≥ n, then the objects Tm(N) and Tm(L) are initial. Since the kernel of a morphism of an object M to an initial object is isomorphic to M, it follows that Tm(M) is an initial object.

(ii) Let N −→ M −→ L be a conﬂation in (CX , EX ) such that M ∈ ObCYn and

L ∈ ObCYn+1 . Then N is an object of CYn . In fact, we have an ’exact’ sequence which yields the ’exact’ sequence z −→ Tm(N) −→ z for all m ≥ n, where z is an initial object of the category CZ . Therefore, Tm(N) is an initial object for m ≥ n.

(iii) This shows that the subcategory CYn of the right exact category (CYn+1 , EYn+1 ) satisﬁes the condition (a) of 7.8.1. The condition (b) of 7.8.1 holds, because CY = CY1 ⊆

CYn and, by hypothesis, for every M ∈ ObCX , there exists a deﬂation P −→ M with P ∈

ObCY . Applying 7.8.1, we obtain that the natural map K•(Yn, EYn ) −→ K•(Yn+1, EYn+1 ) [ is an isomorphism for all n ≥ 1. Since, by hypothesis, CX = CYn , the isomorphisms n≥1 ∼ K•(Yn, EYn ) −→ K•(Yn+1, EYn+1 ) imply that the natural map K0(Y, EY ) −→ K0(X, EX ) is an isomorphism.

92 7.8.3. Proposition. Let (CX , EX ) be a right exact category with initial objects; and let 0 0 β1 α1 Ker(f 0) −−−→ Ker(f) −−−→ Ker(f 00)    0   00 k y k y y k β α 1 1 00 (3) Ker(α1) −−−→ A1 −−−→ A   1 0   00 f y f y y f β α 2 2 00 Ker(α2) −−−→ A2 −−−→ A2 be a commutative diagram (determined by its lower right square) such that Ker(k00) and Ker(β2) are trivial. Then 0 0 (a) The upper row of (3) is ’exact’, and the morphism β1 is the kernel of α1. 0 (b) Suppose, in addition, that the arrows f , α1 and α2 in (3) are deﬂations and (CX , EX ) has the following property:

p (#) If M −→e N is a deﬂation and M −→ M an idempotent morphism (i.e. p2 = p) which has a kernel and such that the composition e ◦ p is a trivial morphism, then the k(p) composition of the canonical morphism Ker(p) −−−→ M and M −→e N is a deﬂation.

Then the upper row of (3) is a conﬂation.

Proof. (a) It follows from C1.5.1 that the upper row of (3) is ’exact’. It follows from 0 β1 the argument of C1.5.1 that the morphism Ker(f 0) −−−→ Ker(f) is the kernel morphism 0 α1 of Ker(f) −−−→ Ker(f 00). (b) The following argument is an appropriate modiﬁcation of the proof the ’snake’ lemma C1.5.2. (b1) We have a commutative diagram

id α 00 e1 00 Ae1 −−−→ Ker(f α1) −−−→ Ker(f )     00  00 id y ek y cart y k ψ0 k00 α e 1 00 (4) Ker(α1) −−−→ Ker(α2f) −−−→ A1 −−−→ A    1     00 id y h y cart f y y f f 0 β α 2 2 00 Ker(α1) −−−→ Ker(α2) −−−→ A2 −−−→ A2 with cartesian squares as indicated. It follows (from the left lower cartesian square of (4)) that Ker(h) is naturally isomorphic to Ker(f).

93 (b2) Since the upper right square of (4) is cartesian, we have a commutative diagram

k(α1) α e 00 e1 00 Ker(α1) −−−→ Ker(f α1) = Ae1 −−−→ Ker(f )  e    00  00 id y ek y cart y k β α 1 1 00 (5) Ker(α1) −−−→ A1 −−−→ A   1 0   00 f y f y y f β α 2 2 00 Ker(α2) −−−→ A2 −−−→ A2

f 0 (b3) Since Ker(α1) −→ Ker(α2) is a deflation, there exists a cartesian square γ M −−−→ Ae1     p y cart y h (6) f 0 Ker(α1) −−−→ Ker(α2) whose upper horizontal arrow, γ, is also a deflation. The commutative diagram (5) shows, among other things, that the arrow f 0 factors s through h (see the diagram (4)), there exists a splitting, Ker(α1) −→ M, of the morphism p p. Set p = s ◦ p. The morphism M −→ M is an idempotent which has the same kernel as p, because s is a monomorphism. ϕ 00 γ (b4) Let M −→ Ker(f ) denote the composition of the deflations M −→ Ae1 and α1 00 Ae1 −→e Ker(f ). The composition ϕ ◦ p is trivial. In fact, ϕ ◦ p = αe1 ◦ γ ◦ s ◦ p, and, by the origin of the morphism s, the composition γ ◦ s coincides with k(αe1); so that ϕ ◦ p = (αe1 ◦ k(αe1)) ◦ p which shows the triviality of ϕ ◦ p. (b5) Suppose that the condition (#) holds. Then the triviality of ϕ ◦ p implies that k(p) the composition ϕ with the canonical morphism Ker(p) −−−→ M is a deflation. It follows from the commutative diagram

id ∼ Ker(p) −−−→ Ker(h) −−−→ Ker(f)       0 y y y α1 γ α e1 00 (7) M −−−→ Ae1 −−−→ Ker(f )     p y cart y h f 0 Ker(α1) −−−→ Ker(α2)

k(p) ϕ that the composition of Ker(p) −−−→ M with M −→ Ker(f 00) equals to the composition α0 of Ker(f) −→1 Ker(f 00) with an isomorphism Ker(p) −→∼ Ker(f). Therefore, the mor- α0 phism Ker(f) −→1 Ker(f 00) is a deﬂation. Together with (a) above, this means that the upper row of the diagram (3) is a conﬂation.

94 7.8.4. Proposition. Let (CX , EX ) be a right exact category with initial objects having the property (#) of 7.8.3. Let CY be a fully exact subcategory of a right exact category (CX , EX ) which has the following properties: (a) If N −→ M −→ L is a conflation in (CX , EX ) and N,M are objects of CY , then L belongs to CY too. (b) For any deflation M −→ L with L ∈ ObCY , there exist a deflation M −→ L with M ∈ ObCY and a morphism M −→ M such that the diagram

M .& M −−−→ L commutes. (c) If P, M are objects of CY and P −→ x is a morphism to initial object, then ` ` P M exists (in CX ) and the sequence P −→ P M −→ M (where the left arrow is the canonical coprojection and the right arrow corresponds to the M −→id M and the composition of P −→ x −→ M) is a conﬂation. Let CY be a full subcategory of CX generated by all objects L having a CY -resolution n S of the length ≤ n. And set CY∞ = n≥0 CYn . Then CYn is a fully exact subcategory of (CX , EX ) for all n ≤ ∞ and the natural morphisms

∼ ∼ ∼ ∼ K•(Y, EY ) −→ K•(Y1, EY1 ) −→ ... −→ K•(Yn, EYn ) −→ K•(Y∞, EY∞ ) are isomorphisms for all n ≥ 0.

Proof. Let N −→ M −→ L be a conﬂation in (CX , EX ). Then for any integer n ≥ 0, we have

(i) If L ∈ ObCYn+1 and M ∈ ObCYn , then N ∈ ObCYn .

(ii) If N and L are objects of CYn+1 , then M is an object of CYn+1 .

(iii) If M and L are objects of CYn+1 , then N is an object of CYn+1 . It suﬃces to prove the assertion for n = 0. 0 0 (i) Since L ∈ ObCY1 , there exists a conﬂation P −→ P −→ L, where P and P are objects of CY . Thus, we have a commutative diagram

id x −−−→ P 0 −−−→ P 0       y y y N −−−→ P −−−→ P (8)  e     id y y cart y N −−−→ M −−−→ L whose rows and columns are conﬂations. Here x is an initial object of the category CX . 0 Since M and P belong to CY and CY is a fully exact subcategory of (CX , EX ), in particular, it is closed under extensions, the object Pe belongs to CY . Since Pe and P are objects of CY , it follows from the condition (a) that N ∈ ObCY .

95 (ii) Since L ∈ ObCY1 , there exists a deflation P −→ L with P ∈ ObCY . Applying (b) to the deflation Pe −→ P in (3), we obtain a deflation M −→ P such that M ∈ ObCY and the composition M −→ L factors through the deflation M −→ L (see (8)). Since 0 N ∈ ObCY1 , there exists a conflation Pe −→ P −→ N where Pe and P are objects of CY . Thus, we obtain a commutative diagram

P0 −−−→ M −−−→ P00 e f e    y y y P −−−→ P ` M −−−→ M (9)       y y y N −−−→ M −−−→ L whose two lower rows and the left and the right columns are conﬂations. By 7.8.3(b), the upper row of (9) is a conﬂation too. Applying (i) to the right column of (9), we obtain 00 that Pe ∈ ObCY . This implies that Mf ∈ ObCY , whence M ∈ ObCY1 .

(iii) Since M ∈ ObCY1 , there is a commutative diagram

id λ P 0 −−−→ P 0 −−−→ x       y y y K −−−→ P −−−→ L (10)  e     y cart y y id N −−−→ M −−−→ L whose rows and columns are conﬂations. Here x is an initial object of CX and λ is a unique morphism P 0 −→ x determined by the fact that P 0 −→ K is the kernel of K −→ N. Since

L ∈ ObCY1 , applying (i) to the middle row, we obtain that K ∈ ObCY . So, N ∈ ObCY .

7.8.5. Proposition. Let (CX , EX ) be a right exact category with initial objects having the property (#) of 7.8.3. Let CY be a fully exact subcategory of a right exact category 0 00 (CX , EX ) satisfying the conditions (a) and (c) of 7.8.4. Let M −→ M −→ M be a 0 0 00 00 conﬂation in (CX , EX ), and let P −→ M , P −→ M be CY -resolutions of the length 00 0 n ≥ 1. Suppose that resolution P −→ M is projective. Then there exists a CY -resolution 0 ` 00 P −→ M of the length n such that Pi = Pi Pi for all i ≥ 1 and the splitting ’exact’ sequence P0 −→ P −→ P00 is an ’exact’ sequence of complexes. Proof. We have the diagram

P0 P00 0 0   y y M 0 −−−→ M −−−→ M 00

00 whose row is a conflation and vertical arrows are deflations. Since, by hypothesis, P0 is 00 a projective object of (CX , EX ) and M −→ M is a deflation, the right vertical arrow,

96 00 00 00 P0 −→ M , factors through M −→ M . Therefore (like in the argument 7.8.4(ii)), we obtain a commutative diagram Ker(e0) −−−→ Ker(e) −−−→ Ker(e00)       y y y P0 −−−→ P0 ` P00 −−−→ P00 0 0  0 0 0    00 e y e y y e M 0 −−−→ M −−−→ M 00 By 7.8.3(b), the upper row of this diagram is a conflation, which allows to repeat the step with the diagram P0 P00 1 1   y y Ker(e0) −−−→ Ker(e) −−−→ Ker(e00) whose vertical arrows are deflations; etc.. 7.9. Characteristic ’exact’ filtrations and sequences.

7.9.1. The right exact ’spaces’ (Xn, EXn ). For a right exact exact ’space’ (X, EX ), let CXn be the category whose objects are sequences Mn −→ Mn−1 −→ ... −→ M0 of n morphisms of EX , n ≥ 1, and morphisms between sequences are commutative diagrams M −−−→ M −−−→ ... −−−→ M n n−1  0    fn y fn−1y ... y f0 0 0 0 Mn −−−→ Mn−1 −−−→ ... −−−→ M0

Notice that if x is an initial object of the category CX , then x −→ ... −→ x is an initial object of CXn .

We denote by EXn the class of all morphisms (fi) of the category CXn such that fi ∈ EX for all 0 ≤ i ≤ n.

7.9.1.1. Proposition. (a) The pair (CXn , EXn ) is a right exact category. (b) The map which assigns to each right exact ’space’ (X, EX ) the right exact ’space’

(Xn, EXn ) extends naturally to an ’exact’ endofunctor of the left exact category (Espr, Les) of right ’exact’ ’spaces’ which induces an ’exact’ endofunctor Pn of its exact subcategory ∗ ∗ (Espr, Les). Proof. The argument is left to the reader.

7.9.2. Proposition. (Additivity of ’characteristic’ ﬁltrations) Let (CX , EX ) and ∗ tn ∗ tn−1 t1 ∗ (CY , EY ) be right exact categories with initial objects and fn −→ fn−1 −→ ... −→ f0 a sequence of deﬂations of ’exact’ functors from (CX , EX ) to (CY , EY ) such that the functors ∗ ∗ X ki = Ker(ti ) are ’exact’ for all 1 ≤ i ≤ n. Then K•(fn) = K•(f0) + K•(ki). 1≤i≤n ∗ Proof. (a) For 1 ≤ i ≤ n, let pY,i denote the functor CYn −→ CY which assigns to γn γ1 every object M = (Mn −→ ... −→ M0) of CYn the object Mi and to every morphism

97 γn γ1 f = (fm) the morphism fi. The assignment to any object M = (Mn −→ ... −→ M0) Y γi ∗ ti ∗ of CYn of CYn the deﬂation Mi −→ Mi−1 is a functor morphism pY,i −→ pY,i−1. Let ∗ Y kY,i denote the kernel of ti , i.e. is the functor CYn −→ CY that assigns to an object γn γ1 γi M = (Mn −→ ... −→ M0) the kernel of Mi −→ Mi−1. Thus, we obtain a diagram

∗ ∗ ∗ kY,n kY,n−1 ... kY,1       y y ... y (1) Y Y ∗ tn ∗ ∗ t1 ∗ pY,n −→ pY,n−1 −→ ... −→ pY,1 −→ pY,0

of functors from CYn to CY whose horizontal arrows are deflations. ∗ ∗ The functors pY,i−1 and kY,i map initial objects to initial objects and pull-backs of deflations to pull-backs of deflations; i.e. they are inverse image functors of morphisms ∗ of the category Espr . These morphisms depend functorially on the right exact ’space’ (Y, EY ), that is they form functor morphisms

pi ki P −−−→ Id ∗ , 0 ≤ i ≤ n, and P −−−→ Id ∗ , 1 ≤ i ≤ n. n Espr n Espr

These morphisms induce morphisms

K•(ki) K•(pi) K• ←−−− K• ◦ Pn −−−→ K•

∗ of ∂ -functors. The claim is that the morphism K•(pn) coincides with the morphism X K•(p0) + K•(ki). 1≤i≤n In fact, the zero components of these morphisms coincide. Since K• is a universal ∂∗-functor, this implies that the entire morphisms coincide with each other. (b) The argument above proves, in a functorial way, the assertion 7.9.2 for the special Y Y ∗ tn ∗ ∗ t1 ∗ case of the sequence of deﬂations pY,n −→ pY,n−1 −→ ... −→ pY,1 −→ pY,0. of ’exact’ functors from CYn to CY . That is

X K•(pY,n) = K•(pY,0) + K•(kY,i). (2) 1≤i≤n

Consider now the general case. ∗ tn ∗ tn−1 t1 ∗ A sequence of deflations fn −→ fn−1 −→ ... −→ f0 of ’exact’ functors from f∗ en (CX , EX ) to (CY , EY ) defines an ’exact’ functor (CX , EX ) −−−→ (CYn , EYn ). The kernels ∗ ki = Ker(ti) map initial objects to initial objects. The fact that they are ’exact’ (which is equivalent to the condition that they map deflations to deflations) means that they ∗ are inverse image functors of morphisms of Espr , hence the morphisms K•(ki) are well

98 X deﬁned. Therefore, the morphism K•(f0) + K•(ki) from K•(X, EX ) to K•(Y, EY ) is 1≤i≤n well deﬁned. One can see that

K•(fn) = K•(pY,n) ◦ K•(efn) and X X K•(f0) + K•(ki) = K•(pY,0) + K•(kY,i) ◦ K•(efn) 1≤i≤n 1≤i≤n So that the assertion follows from the equality (2).

7.9.3. Corollary. Let (CX , EX ) and (CY , EY ) be right exact categories with initial ∗ ∗ ∗ objects and g −→ f −→ h a conflation of ’exact’ functors from (CX , EX ) to (CY , EY ). Then K•(f) = K•(g) + K•(h). 7.9.4. Corollary. (Additivity for ’characteristic’ ’exact’ sequences) Let ∗ ∗ ∗ ∗ fn −→ fn−1 −→ ... −→ f1 −→ f0 be an ’exact’ sequence of ’exact’ functors from (CX , EX ) to (CY , EY ) which map initial ∗ ∗ ∗ ∗ objects to initial objects. Suppose that f1 −→ f0 is a deflation and fn −→ fn−1 is the kernel ∗ ∗ X i of fn−1 −→ fn−2. Then the morphism (−1) K•(fi) from K•(X, EX ) to K•(Y, EY ) is 0≤i≤n equal to zero. Proof. The assertion follows from 7.9.3 by induction. A more conceptual proof goes along the lines of the argument of 7.9.2. Namely, we assign to each right exact category (C , E ) the right exact category (C e , E e ) whose Y Y Yn Yn objects are ’exact’ sequences L = (Ln −→ Ln−1 −→ ... −→ L1 −→ L0), where L1 −→ L0 is a deflation and Ln −→ Ln−1 is the kernel of Ln−1 −→ Ln−2. This assignment defines e ∗ an endofunctor Pn of the category Espr of right exact ’spaces’ with initial objects, and e maps L 7−→ L determine morphisms P −→ Id ∗ . The rest of the argument is left to i n Espr the reader. 7.10. Complements.

7.10.1. Another description of the functor K0. Fix a right exact category

(CX , EX ). Let CL(X,EX ) denote the category having the same objects as CX and with morphisms defined as follows. For any pair M,L of objects, consider all diagrams (if e f any) of the form M ←− Mf −→ L, where e is a deflation and f an arbitrary morphism of CX . We consider isomorphisms between such diagrams of the form (idM , φ, idL) and define morphisms from M to L as isomorphism classes of these diagrams. The composition t g e f of the morphisms N ←− Ne −→ M and M ←− Mf −→ L is the morphism represented by 0 the pair (t ◦ e, f ◦ g ) in the diagram g0 f N −−−→ M −−−→ L e f   e y cart y e g N −−−→ M e  t y M

99 with cartesian square. If the category CX is svelte (i.e. it represents a ’space’), then

CL(X,EX ) is a well deﬁned svelte category. ∗ rX There is a canonical functor CX −→ CL(X,EX ) which is identical on objects and maps f f each morphism M −→ L to the morphism represented by the diagram M ←−id M −→ L.

Let C|EX | denote the subcategory of CX formed by all deﬂations. The map which e assigns to every morphism M −→ N of EX the morphism of CL(X,EX ) represented by the l∗ diagram N ←−e M −→id M is a functor Cop −→X C . |EX | L(X,EX ) Let G(X) denote the group Z|CX | which is identiﬁed with the corresponding groupoid with one object. Let pX denote the map HomCL(X,EX ) −→ G(X) which assigns to a f f morphism [N ←−e M −→ L] represented by the diagram N ←−e M −→ L the element [M] − [N] of the group G(X). We have a (non-commutative) diagram

p ×p 2 X X Hom CL(X,E ) −−−→ G(X) × G(X)  X    c y y + (1) pX

HomCL(X,EX ) −−−→ G(X)

2 where Hom CZ stands for the class of composable morphisms of the category CZ and the vertical arrows are compositions. Taking the compositions in the diagram (1), we obtain a pair of arrows uX 2 −→ Hom CL(X,EX ) −→ G(X). (2) vX

7.10.1.1. Proposition. The cokernel of the pair (2) is (isomorphic to) the group K0(X, EX ) deﬁned in 7.1. Proof. The fact follows from the deﬁnitions.

pX 7.10.1.2. Note. The map HomCL(X,EX ) −→ G(X) is the composition of the map πX λX HomCL(X,EX ) −→ EX and the map EX −→ G(X) which assigns to each deﬂation M −→ L ∗ the element [M] − [L] of G(X). One can see that πX ◦ lX is the identical map, and the op map λX is a functor C −→ G(X). |EX |

f∗ 7.10.1.3. Functorialities. Any ’exact’ functor (CX , EX ) −→ (CY , EY ) between L(f)∗ right exact categories induces a functor CL(X,EX ) −−−→ CL(Y,EY ) such that the diagram

l∗ r∗ λX op X X K0(X) ←−−− G(X) ←−−− C|E | −−−→ CL(X,EX ) ←−−− CX   X      ∗   ∗ K0(f) y G(f) y y L(f) y y f (3) l∗ r∗ λY op Y Y K0(Y ) ←−−− G(Y ) ←−−− C −−−→ C ←−−− CY |EY | L(Y,EY )

100 commutes, as well as the diagram

πX HomCL(X,E ) −−−→ EX  X  ∗   ∗ f y y E(f) (4) πY HomCL(Y,EY ) −−−→ EY

7.10.2. The Q-construction for right exact categories with initial objects. Let (CX , EX ) be a right exact category with initial objects. We denote by IX the class of ∞ all inﬂations of (CX , EX ) (i.e. morphisms which are kernels of deﬂations) and by IX the smallest subcategory of CX containing IX .

We denote by CQ(X,EX ) the subcategory of the category CL(X,EX ) formed by all e j ∞ morphisms M ←− Mf −→ L, where (e is a deﬂation and) j ∈ IX . ∞ 7.10.2.1. Note. If (CX , EX ) is an exact k-linear category, then IX = IX and the category CQ(X,EX ) coincides with the Quillen’s category QCX associated with the exact category (CX , EX ) (see [Q, p. 102]). Let aX 2 −−−→ Hom CQ(X,EX ) −−−→ G(X). (1) bX be the composition of the pair of maps 7.10.1(2) with the embedding

2 2 Hom CQ(X,EX ) −−−→ Hom CL(X,EX ).

7.10.2.2. Proposition. The unique map Cok(aX , bX ) −→ K0(X, EX ) making commute the diagram

aX 2 −−−→ Hom CQ(X,EX ) −−−→ G(X) −−−→ Cok(aX , bX ) b  X      y id y y uX Hom2C −−−→ G(X) −−−→ K (X, E ) L(X,EX ) −−−→ 0 X vX is a group isomorphism. Proof. The assertion is a consequence of 7.1.6.

101 8. Infinitesimal ’spaces’. 8.1. The Gabriel multiplication in right exact categories. Fix a right exact category (CX , EX ) with initial objects. Let T and S be subcategories of the category CX . The Gabriel product S•T is the full subcategory of CX whose objects M fit into a conflation g h L −→ M −→ N such that L ∈ ObS and N ∈ ObT.

8.1.1. Proposition. Let (CX , EX ) be a right exact category with initial objects. For any subcategories A, B, and D of the category CX , there is the inclusion

A• (B•D) ⊆ (A•B) •D.

Proof. Let A, B, and D be subcategories of CX . Let M be an object of A• (B•D); i.e. there is a conﬂation L −→ M −→ N such that L ∈ ObA and N ∈ ObB•D. The latter means that there is a conﬂation N1 −→ N −→ N2 with N1 ∈ ObB and N2 ∈ ObD. Thus, we have a commutative diagram

L −−−→ M −−−→ N  1 1    id y y cart y L −−−→ M −−−→ N     y y id N2 −−−→ N2 whose two upper right square is cartesian, and two upper rows and two right columns are conﬂations. So, we have a conﬂation M1 −→ M −→ N2 with N2 ∈ ObD and M1 ∈ ObA•B, hence M is an object of the subcategory (A•B) •D.

8.1.2. Corollary. Let (CX , EX ) be an exact category. Then the Gabriel multiplication is associative.

Proof. Let A, B, and D be subcategories of CX . By 8.1.1, we have the inclusion A• (B•D) ⊆ (A•B) •D. The opposite inclusion holds by duality, because (A•B)op = Bop •Aop.

8.2. The inﬁnitesimal neighborhoods of a subcategory. Let (CX , EX ) be a right exact category with initial objects. We denote by OX the full subcategory of CX generated by all initial objects of CX . For any subcategory B of CX , we deﬁne subcategories (n) (0) (1) B and B(n), 0 ≤ n ≤ ∞, by setting B = OX = B(0), B = B = B(1), and [ B(n) = B(n−1) •B for 2 ≤ n < ∞; and B(∞) = B(n); n≥1 [ B(n) = B•B(n−1) for 2 ≤ n < ∞; and B(∞) = B(n) n≥1

(n) (n) It follows that B = B(n) for n ≤ 2 and, by 8.1.1, B(n) ⊆ B for 3 ≤ n ≤ ∞.

102 We call the subcategory B(n+1) the upper nth inﬁnitesimal neighborhood of B and the th (n+1) subcategory B(n+1) the lower n inﬁnitesimal neighborhood of B. It follows that B is the strictly full subcategory of CX generated by all M ∈ ObCX such that there exists a sequence of arrows

j1 j2 jn M0 −−−→ M1 −−−→ ... −−−→ Mn = M

ei with the property: M0 ∈ ObB, and for each n ≥ i ≥ 1, there exists a deflation Mi −→ Ni ji ei with Ni ∈ ObB such that Mi−1 −→ Mi −→ Ni is a conflation. Similarly, B(n+1) is a strictly full subcategory of CX generated by all M ∈ ObCX such that there exists a sequence of deflations

en e2 e1 M = Mn −−−→ ... −−−→ M1 −−−→ M0 such that M0 and Ker(ei) are objects of B for 1 ≤ i ≤ n. 8.2.1. Note. It follows that B(n) ⊆ B(n+1) for all n ≥ 0, if B contains an initial object of the category CX . 8.3. Fully exact subcategories of a right exact category. Fix a right exact category (CX , EX ). A subcategory A of CX is a fully exact subcategory of (CX , EX ) if A•A = A.

8.3.1. Proposition. Let (CX , EX ) be a right exact category with initial objects. For (∞) any subcategory B of CX , the subcategory B is the smallest fully exact subcategory of (CX , EX ) containing B.

Proof. Let A be a fully exact subcategory of the right exact category (CX , EX ), i.e. A = A•A. Then B(∞) ⊆ A, iﬀ B is a subcategory of A. On the other hand, it follows from 8.1.1 and the deﬁnition of the subcategories B(n) (see 8.2) that B(n) •B(m) ⊆ B(m+n) for any nonnegative integers n and m. In particular, (∞) (∞) (∞) (∞) B = B •B , that is B is a fully exact subcategory of (CX , EX ) containing B.

8.4. Cofiltrations. Fix a right exact category (CX , EX ) with initial objects. A cofiltration of the length n+1 of an object M is a sequence of deflations

en e2 e1 M = Mn −−−→ ... −−−→ M1 −−−→ M0. (1)

The coﬁltration (1) is said to be equivalent to a coﬁltration

een ee2 ee1 M = Mfm −−−→ ... −−−→ Mf1 −−−→ Mf0 if m = n and there exists a permutation σ of {0, . . . , n} such that Ker(ei) ' Ker(eσ(i)) for 1 ≤ i ≤ n and M0 ' Mf0. The following assertion is a version (and a generalization) of Zassenhouse’s lemma.

8.4.1. Proposition. Let (CX , EX ) have the following property:

103 t1 t2 (‡) for any pair of deﬂations M1 ←− M −→ M2, there is a commutative square

t1 M −−−→ M  1   t2 y y p2 p1 M2 −−−→ M3

of deflations such that the unique morphism M −→ M1 ×M3 M2 is a deflation. Then any two cofiltrations of an object M have equivalent refinements. Proof. Let

en e2 e1 M = Mn −−−→ ... −−−→ M1 −−−→ M0 and

een ee2 ee1 M = Mfm −−−→ ... −−−→ Mf1 −−−→ Mf0 be cofiltrations. If n = 0, then the second cofiltration is a refinement of the first one.

e1 e1 (a) Suppose that n = 1 = m; that is we have a pair of deﬂations Mf1 ←−e M −→ M1. Thanks to the property (‡), there exists a commutative square

e1 M −−−→ M  1   e1 y y p1 0 p1 Mf1 −−−→ N

e3 whose all arrows are deﬂations, and the unique arrow M −→ M2 = M1 ×N Mf1 is a deﬂation too. Since the right lower square in the commutative diagram

∼ Ker(e2) −−−→ Ker(p1) e    ek2 y y k1 k2 e2 Ker(e2) −−−→ M2 −−−→ M1       oy e2 y cart y p1 k0 p0 0 1 1 Ker(p1) −−−→ Mf1 −−−→ N is cartesian, its upper horizontal and left vertical arrows are isomorphisms. This shows that the coﬁltrations

e3 e2 p1 M −−−→ M2 −−−→ M1 −−−→ N and p0 e3 ee2 1 M −−−→ M2 −−−→ Mf1 −−−→ N are equivalent to each other.

104 e1 en (b) Let n > 1 and m = 1. Then, applying (a) to the deﬂations Mf0 ←−e M −→ Mn−1, we obtain a commutative diagram

0 e en en M −−−→ M 0 −−−→ M −−−→ M −−−→ ... −−−→ M  n−1 n−2 0   e1 y cart y p1 0 p1 Mf1 −−−→ N which provides an induction argument. (c) Finally, (b) provides the main induction step in the general case. Details are left to the reader. 8.5. Semitopologizing, topologizing, and thick subcategories of a right exact category. Fix a right exact category (CX , EX ) with initial objects.

8.5.1. Deﬁnitions. (a) We call a full subcategory T of the category CX semitopologizing if the following condition holds: e If M −→ L is an arrow of EX and M ∈ ObT , then L and Ker(e) are objects of T . (b) We call a semitopologizing subcategory T of the category CX topologizing if it is a right exact subcategory of (CX , EX ), that is if

f 0 N −−−→ M e    e y y e f N −−−→ L is a cartesian square in CX and the objects M, L, and N belong to the subcategory T , then Ne is an object of T . (c) We call a subcategory T of CX a thick subcategory of (CX , EX ) if it is topologizing and fully exact, i.e. T•T = T .

8.5.2. Proposition. (a) Let (CX , EX ) be a right exact category with initial and final objects such that all morphisms to final objects are deflations. Than any topologizing subcategory of (CX , EX ) is closed under finite products. (b) If CX is an abelian category and EX is the canonical exact structure on CX , then topologizing subcategories of (CX , EX ) are topologizing subcategories of the abelian category CX in the sense of Gabriel [Gab].

Proof. (a) Let T be a topologizing subcategory of (CX , EX ). Then, under the assumptions, it is a strictly full subcategory of CX containing all final objects of CX . Let M,N be objects of T and x• a final object of CX . By hypothesis, the unique morphisms M −→ x• and N −→ x• are deflations. Therefore, the cartesian square

pM M Q N −−−→ M   p   N y y N −−−→ x•

105 is contained in T . (b) If (CX , EX ) is an abelian category with the canonical structure, then it follows from (a) above that any topologizing subcategory T of (CX , EX ) is closed under ﬁnite (co)products, and if 0 −→ M 0 −→ M −→ M 00 −→ 0 is an exact sequence with M ∈ ObT , then M 0 and M 00 are objects of T . This means that T is a topologizing subcategory of the abelian category CX in the sense of Gabriel. On the other hand any topologizing subcategory of CX in the sense of Gabriel is closed under any ﬁnite limits and colimits (taken in CX ), in particular, it is closed under arbitrary pull-backs.

8.5.3. Proposition. Let (CX , EX ) be a k-linear additive right exact category such that all morphisms to zero objects are deflations. (a) Any topologizing subcategory of (CX , EX ) is closed under finite products. (b) If (CX , EX ) is an exact category, then any topologizing subcategory of (CX , EX ) is an exact (sub)category. Proof. (a) This follows from 8.5.2(a). (b) Fix a topologizing subcategory T of an exact k-linear category (CX , EX ). Let f M −→j M 0 −→e M 00 be a conflation in T and M −→ L an arbitrary morphism of T . Since (CX , EX ) is an exact category, there is cocartesian square

j M −−−→ M 0     0 f y y f (1) j N −−−→e N 0 whose horizontal arrows are inﬂations. Notice that the pair of morphisms

(f,j) j+f 0 M −−−→ N × M 0 = N ⊕ M 0 −−−→e N 0 (2) is a conflation. In fact, the Gabriel-Quillen embedding is ’exact’, hence it sends the cocartesian square (1) to a cocartesian square of the abelian category of sheaves of k-modules on (CX , EX ). And for abelian categories the fact is easy to check. Since the Gabriel-Quillen embedding reflects conflations, it follows that (2) is a conflation. By (a) above, N ⊕ M 0 ∈ ObT , because N and M 0 are objects of T . Therefore, the object N 0 belongs to T .

f ∗ 8.5.4. Proposition. Let (X, EX ) −→ (Y, EY ) be a morphism of the category Espr . If T is a semitopologizing (resp. topologizing, resp. thick) subcategory of the right exact ∗−1 category (CX , EX ), then f (T ) is a semitopologizing (resp. topologizing, resp. thick) subcategory of (CY , EY ).

∗ ∗ Proof. By the definition of morphisms of Espr , the inverse image functor f is an ’exact’ (that is preserving pull-backs of deflations) functor from (CY , EY ) to (CX , EX ) which maps initial objects to initial objects. The assertion follows from definitions.

106 8.5.5. Proposition. Let

g (Z, EZ ) −−−→ (Y, EY )     f y y p1 p2 (X, EX ) −−−→ (X, EX)

∗ be a cocartesian square in the category Espr , and let CX0 ,CY0 be semitopologizing subcat- Y egories of resp. (CX , EX ) and (CY , EY ). Then CX0 = CX0 CY0 is a semitopologizing ∗ ∗ f0 ,g0 subcategory of (CX, EX). If the subcategories CX0 and CY0 are topologizing, then CX0 is a topologizing subcategory of (CX, EX).

(ξ,γ) 0 0 0 Proof. (a) By 6.8.2, EX consists of all morphisms (M,L; φ) −−−→ (M ,L ; φ ) of the 00 category CX such that ξ ∈ EX and γ ∈ EY . And Ker(ξ, γ) = (Ker(ξ), Ker(γ); φ ), where φ00 is a uniquely determined (once Ker(ξ) and Ker(γ) are ﬁxed) isomorphism. Therefore, if (M,L; φ) is an object of CX0 and both categories CX0 and CY0 are semitopological, then 0 0 0 (M ,L ; φ ) and Ker(ξ, γ) are objects of CX0 , which shows that CX0 is a semitopological subcategory of the category CX.

(b) Suppose now that CX0 and CY0 are topologizing subcategories of respectively ∗ (CX , EX ) and (CY , EY ). By definition of morphisms of the category Espr , the inverse f ∗ g∗ image functors CX −→ CZ and CY −→ CZ are ’exact’; i.e. they preserve pull-backs of (ξ,γ) deflations. This implies that for any deflation (M,L; φ) −−−→ (M 0,L0; φ0) and an arbitary (α,β) 00 00 00 0 0 0 morphism (M ,L ; φ ) −−−→ (M ,L ; φ ) of the category CX, there exists a cartesian square 0 (p2,p2) (M,f Le; φe) −−−→ (M,L; φ)   0   (p1, p1) y y (ξ, γ) (α,β) (M 00,L00; φ00) −−−→ (M 0,L0; φ0) determined uniquely up to isomorphism by the fact that the squares

0 p2 p2 L −−−→ L M −−−→ M e  f    0   p1 y y ξ and p1 y y γ ξ γ L00 −−−→ L0 M 00 −−−→ M 0

00 are both cartesian. Therefore, if L and L are objects of the topologizing subcategory CY0 , 00 then Le ∈ ObCY0 . Similarly, Mf ∈ ObCX0 if M and M are objects of CX0 . This shows that CX0 is a topologizing subcategory of (CX, EX).

8.6. Another left exact structure Lesp on the category Espr of right exact q ’spaces’. We denote by Lesp the class of all morphisms (X, EX ) −→ (Y, EY ) of right exact

107 0 ∗ ∗ e ∗ 0 ’spaces’ such that q is a localization functor and for each arrow q (L) −→ q (L ) of EX , e 00 ∗ 00 s ∗ 0 there exists an arrow L −→ L of EY and an isomorphism q (L ) −→ q (L ) such that e0 = s ◦ q∗(e). It follows from this deﬁnition that the class Lesp is contained in Les.

8.6.1. Proposition. The class Lesp is a left exact structure on the category Espr of right exact ’spaces’.

Proof. The class Lesp contains, obviously, all isomorphisms, and it is easy to see that it is closed under composition. It remains to show that Lesp is stable under cobase change and its arrows are cocovers of a subcanonical copretopology. q f Let (X, EX ) −→ (Y, EY ) be a morphism of Lesp and (X, EX ) −→ (Z, EZ ) an arbitrary q a morphism. The claim is that the canonical morphism Z −→e Z Y belongs to Lesp. f,q Consider the corresponding cartesian (in pseudo-categorical sense) square of right exact categories: p∗ (CX, EX) −−−→ (CY , EY )   ∗   ∗ qe y y q (2) f ∗ (CZ , EZ ) −−−→ (CX , EX )

a Y ∗ where X = Z Y ; that is CX = CZ CY . Recall that the functor qe maps each object f,q f ∗,q∗ (L, M; φ) of the category CX to the object L of CZ and each morphism (ξ, γ) to ξ. By ∗ ∗ 6.1(a), qe is a localization functor (because q is a localization functor). 0 0 0 Let (L, M; φ) and (L ,M ; φ ) be objects of the category CX; and let

0 ∗ e 0 ∗ 0 0 0 qe (L, M; φ) = L −→ L = qe (L ,M ; φ ) (3)

f ∗(e) ∗ ∗ 0 be an arrow of EZ . Then f (L) −−−→ f (L ) is a morphism of EX . Since the localization q e 00 X −→ Y belongs to Lesp, there exists a morphism M −→ M of EY and an isomorphism ψ q∗(M 0) −→ q∗(M 00) such that the diagram

f ∗(e) f ∗(L) −−−→ f ∗(L0)     0 φ yo oy ψ ◦ φ q∗(e) q∗(M) −−−→ q∗(M 00) commutes. This expresses the fact that the pair (e0, e) is a morphism from the object 0 00 0 (L, M; φ) to the object (L ,M ; ψφ ) of the category CX. The morphism

(e,e0) (L, M; φ) −−−→ (L0,M 00; ψφ0)

108 0 ∗ 0 0 is a deﬂation, because both e and e are deﬂations. Finally, qe (e , e) = e . This shows that the class of localizations Lesp is stable under cobase change; i.e. Lesp is the class of cocovers of a copretopology. Since the copretopology Lesp is coarser than Les and Les is subcanonical, it follows that Lesp is subcanonical too. In other words, Lesp is a left exact structure on the category Espr of right exact ’spaces’. q 8.6.2. Proposition. Let (X, EX ) −→ (Y, EY ) be a morphism of Les satisfying the following conditions: s e (i) Every pair of arrows N ←− M −→ L, where e ∈ EY and s ∈ Σq∗ = {t ∈ ∗ HomCY | q (t) is invertible}, can be completed to a commutative square

e M −−−→ L     s y y t e0 N −−−→ L0

0 with e ∈ EY and t ∈ Σq∗ . s e (ii) Every pair of arrows N −→ M −→ L, where e ∈ EY and s ∈ Σq∗ , can be completed to a commutative square e M −−−→ L x x   s   t e0 N −−−→ L0

0 with e ∈ EY and t ∈ Σq∗ .

Then q belongs to the class Lesp.

0 ∗ e ∗ Proof. Since q ∈ Les, for each morphism q (M) −→ q (L), there exists a commutative diagram e0 q∗(M) −−−→ q∗(L)     oy yo q∗(t) q∗(Mf) −−−→ q∗(Le)

∗ ∼ ∗ whose vertical arrows are isomorphisms and t ∈ EY . Any isomorphism q (M) −→ q (Mf) is ∗ ±1 the composition of morphisms q (s) , where s ∈ Σq∗ . Therefore, applying the conditions (i) and (ii), we obtain, in a ﬁnite number of steps, a commutative diagram of the form

e0 q∗(M) −−−→ q∗(L)     id y yo q∗(t0) q∗(M) −−−→ q∗(L0)

0 in which t ∈ EX.

109 8.6.3. Proposition. Let CX be a category with kernel pairs and finite colimits, whose st canonical (i.e. the finest) right exact structure EX consists of all strict epimorphisms (e.g. CX is a quasi-abelian category). Then any right exact and ’exact’ localization functor ∗ q st q st CX −→ CZ is an inverse image functor of a morphism (Z, EZ ) −→ (X, EX ) which belongs to Lesp. f Proof. It follows that each morphism L −→ M is the composition of a strict epimor- tf jf phism L −→ Me (– the cokernel of the kernel pair of f) and a monomorphism Me −→ M. ∗ Being right exact, the localization functor q maps the strict epimorphism tf to (the cokernel of a pair of arrows, hence) a strict epimorphism and the monomorphism jf to an isomorphism. This implies, in particular, the condition (ii). Since the category CX has ∗ finite colimits and the localization functor q is right exact, the class of morphisms Σq∗ is a left multiplicative system [GZ, 8.3.4]. In particular, the condition (i) holds.

8.6.4. Note. One can show that the quotient category CZ satisfies the same property: the class of strict epimorphisms is stable under base change. ∗ ∗ ∗ 8.6.5. The left exact structure Lesp on the category Espr . We denote by Lesp T ∗ the intersection Lesp Les. ? 8.7. The K-functor K•. Applying 8.4.2.1 to the identical functor ∗ ∗ op ∗ ∗ op (Espr , Lesp) −−−→ (Espr , Les) ∗ ∗ (i.e. restricting to a coarser left exact structure Lesp) we obtain the universal ∂ -functor ? ? ? ∗ ∗ op K• = (Ki , di | i ≥ 0) from (Espr , Lesp) to Z − mod whose zero component coincides with K∗ ∗ op 0 the functor (Espr ) −−−→ Z − mod. 8.8. The left exact category of right exact infinitesimal ’spaces’. We define a right exact infinitesimal ’space’ as a pair ((X, EX ),Y ), where (CX , EX ) is a right exact category with initial objects and CY is a topologizing subcategory of (CX , EX ) such that CX = (CY )(∞). A morphism ((X, EX ),Y ) −→ ((X, EX), Y) of right exact infinitesimal f ’spaces’ is given by a morphism X −→ X of ’spaces’ whose inverse image functor is ’exact’, maps initial objects and the subcategory CY to the subcategory CY . The composition of morphisms is given by the composition of the corresponding morphisms of ’spaces’. This r defines the category which we denote by Esp∞. It follows from this definition that the maps

((X, EX ),Y ) 7−→ (X, EX ) and ((X, EX ),Y ) 7−→ (Y, EY )

(where EY is the induced right exact structure on CY ) extend naturally to functors respectively F! F∗ r ∗ r ∗ Esp∞ −−−→ Espr and Esp∞ −−−→ Espr .

F ∗ ∗ r 8.8.1. Proposition. The functor Espr −−−→ Esp∞ which assigns to each right exact ’space’ (X, EX ) the corresponding ’trivial’ inﬁnitesimal ’space’, ((X, EX ),X) and acts accordingly on morphisms is left adjoint to the functor F! and right adjoint to F∗.

110 ε! ! F Proof. The adjunction morphism F ◦ F −−−→ Id r assigns to each object ∗ Esp∞ idX ((X, EX ),Y ) the morphism ((X, EX ),X) −−−→ ((X, EX ),Y ). The adjunction arrow ! Id ∗ −−−→ F ◦ F is the identical morphism. Espr ∗ η F ∗ The adjunction morphism Id r −−−→ F F assigns to each object ((X, E ),Y ) Esp∞ ∗ X j the morphism ((X, EX ),Y ) −−−→ ((Y, EX ),Y ), where j = jY is the morphism X −→ Y whose inverse image functor is the inclusion functor CY −→ CX . The adjunction arrow ∗ F F −→ Id ∗ is identical. ∗ Espr

8.8.2. Note. Thanks to the full faithfulness of the functor F∗, there is a canonical ρF morphism F! −−−→ F∗ deﬁned as the composition of

−1 F!η η! F∗ ! F ! ∗ ! ∗ F ∗ F −−−→ F F∗F and F F∗F −−−→ F .

The morphism ρF assigns to each inﬁnitesimal right exact ’space’ ((X, EX ),Y ) the j natural morphism (X, EX ) −→ (Y, EY ) whose inverse image functor is the embedding CY −→ CX . ∞ ∞ 8.8.3. The class of morphisms Lesp. The class Lesp consists of all morphisms q q ∗ ((X, EX ),X0) −−−→ ((Y, EY ),Y0) such that (X, EX ) −−−→ (Y, EY ) belongs to Lesp (in ∗ q ∗−1 particular, CY −→ CX is an ’exact’ localization) and CY0 = q (CX0 ). q ∞ 8.8.3.1. Note. For any morphism ((X, EX ),X0) −−−→ ((Y, EY ),Y0) of Lesp, we have ∗ the inclusion Ker(q ) ⊆ CY0 .

In fact, any semitopologizing subcategory of CX , in particular CX0 , contains the initial objects of CX (they are, for instance, kernels of identical morphisms). Therefore, ∗−1 ∗ the equality CY0 = q (CX0 ) implies that Ker(q ) ⊆ CY0 . ∞ 8.8.4. Proposition. The class of morphisms Lesp is a left exact structure on the r category Esp∞ of right exact inﬁnitesimal ’spaces’. ! ∗ r ∞ The functors F and F are ’exact’ functors from the left exact category (Esp∞, Lesp) ∗ ∗ to the left exact category (Espr , Lesp). f q Proof. (a) Let ((X, EX ),X0) ←− ((Z, EZ ),Z0) −→ ((Y, EY ),Y0) be a pair of mor- r q phisms of the category Esp∞. Suppose that the morphism (Z, EZ ) −→ (Y, EY ) belongs to ∗ Lesp. Then there exists a canonical cocartesian square q ((Z, EZ ),Z0) −−−→ ((Y, EY ),Y0)     f y y p1 p2 ((X, EX ),X0) −−−→ ((X, EX), X0)

a a f0 q0 where (X, EX) = (X, EX ) (Y, EY ) and X0 = X0 Y0. Here X0 ←− Z0 −→ Y0 are

f,q f0,q0 morphisms induced by resp. f and q.

111 It follows from 8.5.5 that CX0 is a topologizing subcategory of the right exact category

(CX.EX). The claim is that, under the assumptions, (CX0 )(∞) = CX; i.e. ((X, EX), X0) r is an object of the category Esp∞. In other words, we need to show that each object Y (M,L; φ) of the category CX = CX CY has a CX0 -cofiltration. f ∗,q∗ en e2 e1 Let M = Mn −→ ... −→ M1 −→ M0 be a CX0 -cofiltration of the object M. ∗ The functor f maps this cofiltration to a CZ0 -cofiltration

f ∗(e ) f ∗(e ) f ∗(e ) f ∗(e ) ∗ ∗ n ∗ n−1 2 ∗ 1 ∗ f (M) = f (Mn) −−−→ f (Mn−1) −−−→ ... −−−→ f (M1) −−−→ f (M0)

∗ q of the object f (M). Since (Z, EZ ) −→ (Y, EY ) is a morphism of Lesp and we have an ∗ φ ∗ tn isomorphism f (M) −→ q (L), there exists a deﬂation L −→ Ln−1 and an isomorphism φn−1 ∗ ∗ f (Mn−1) −−−→ q (Ln−1) such that the diagram

f ∗(e ) ∗ n ∗ f (M) −−−→ f (Mn−1)     φ yo oy φn−1 q∗(t ) ∗ n ∗ q (L) −−−→ q (Ln−1) commutes. Continuing this process, we obtain a commutative diagram

∗ ∗ ∗ ∗ f (e ) f (en−1) f (e ) f (e ) ∗ n ∗ 2 ∗ 1 ∗ f (M) −−−→ f (Mn−1) −−−→ ... −−−→ f (M1) −−−→ f (M0)         φ yo oy φn−1 oy φ1 oy φ0 (1) ∗ ∗ ∗ ∗ q (t ) q (tn−1) q (t ) q (t ) ∗ n ∗ 2 ∗ 1 ∗ q (L) −−−→ q (Ln−1) −−−→ ... −−−→ q (L1) −−−→ q (L0) whose vertical arrows are isomorphisms and horizontal arrows are images of deﬂations. Therefore, the diagram (1) encodes a coﬁltration

(en,tn) (en−1,tn−1) (e2,t2) (e1,t1) (M,L; φ) −−−→ (Mn−1,Ln−1; φn−1) −−−→ ... −−−→ (M1,L1; φ1) −−−→ (M0,L0; φ0)

of the object (M,L; φ). This coﬁltration is a CX0 -coﬁltration. (em,tm) In fact, the kernel of the morphism (Mm,Lm; φm) −−−→ (Mm−1,Lm−1; φm−1) is ∗ ψm ∗ isomorphic to (Ker(em), Ker(tm); ψm), where f (Ker(em)) −→ q (Ker(tm)) is a uniquely determined isomorphism. The kernel Ker(em) is an object of CX0 by choice, and Ker(tm) ∗ ∗−1 is an object of CY0 , because q (Ker(tm)) ∈ ObCZ0 and CY0 = q (CZ0 ), because q is a ∞ morphism of Lesp (cf. 8.8.3). r (b) It follows from the description of cocartesian squares in the category Esp∞ (given ! ∞ ! above) that the functor F preserves push-forwards of morphisms of Lesp; that is F is r ∞ an ’exact’ functor from the left exact category (Esp∞, Les ) to the left exact category ∗ ∗ (Espr , Les).

112 ∗ (c) By 8.8.1, the functor F has a right adjoint, F∗, hence it preserves all colimits, in ∗ ∞ ∗ particular, cocartesian squares; and F maps inflations to inflations (that is Lesp to Lesp). ∗ r ∞ ∗ ∗ Therefore, F is an ’exact’ functor from (Esp∞, Les ) to (Espr , Les). 8.9. Right exact infinitesimal ’spaces’ and derived functors. By 8.8.4, the canonical functors F! F∗ ∗ r ∗ Espr ←−−− Esp∞ −−−→ Espr ! ∗ defined by F ((X, EX ),Y ) = (X, EX ) and F ((X, EX ),Y ) = (Y, EY ) (cf. 8.8), are ’exact’ r ∞ ∗ ∗ functors from the left exact category (Esp∞, Lesp) to the left exact category (Espr , Lesp). Let (CZ , EZ ) be a right exact category with initial objects and limits of filtered systems. ∗ op And let G be a functor (Espr ) −→ CZ . Applying Q8.4.2 to G and each of the functors ! ∗ ∗ F! F∗ r ∞ op F and F , we obtain two universal ∂ -functors, G• and G• from (Esp∞, Lesp) to CZ op op whose zero components are respectively the functors G ◦ F! and G ◦ F∗ ; that is

F! F∗ G0 ((X, EX ),Y ) = G(X, EX ) and G0 ((X, EX ),Y ) = G(Y, EY ) for any inﬁnitesimal right exact ’space’ ((X, EX ),Y ). The canonical functor morphism ρ ! F ∗ F −−−→ F (see 8.8.2) assigns to each inﬁnitesimal right exact ’space’ ((X, EX ),Y ) j the morphism (X, EX ) −→ (Y, EY ) whose inverse image functor is the inclusion functor j∗ CY −→ CX , induces a morphism

op GρF op G ◦ F∗ −−−→ G ◦ F! .

∗ ∗ ! Thanks to the universality of the ∂ -functor G• ◦ F , the latter morphism determines uniquely a morphism ρF F∗ • F! G• −−−→ G• ∗ ∗ F! F∗ of universal ∂ -functors. Thanks to the universality of the ∂ -functors G• and G• , there are natural morphisms of ∂∗-functors

ϕ ψ ∗op • F∗ !op • F! G• ◦ F −−−→ G• and G• ◦ F −−−→ G• (1) such that the diagram op G•ρF op G ◦ F∗ −−−→ G ◦ F! •  •   ϕ• y y ψ• (2) ρF F∗ • F! G• −−−→ G• commutes.

8.9.1. Proposition. Let CZ be a category with initial objects and limits of ﬁltered ∗ ∗ op systems; and let G be a functor (Espr , Lesp) −−−→ CZ . Then the natural morphism

!op !op S−G ◦ F −−−→ S−(G ◦ F )

113 is an isomorphism. Here the functor F! is viewed as a morphism from the left exact category r ∞ ∗ ∗ (Esp∞, Lesp) to the left exact category (Espr , Lesp). ψ ? !op • F! ∗ In particular, the morphism K• ◦ F −−−→ K• is an isomorphism of ∂ -functors. r ! Proof. (a) Fix an object ((X, EX ),Y ) of the category Esp∞. The functor F induces ∞ an isomorphism from the category ((X, EX ),Y )\Lesp of inﬂations of ((X, EX ),Y ) onto the ∗ ! category (X, EX )\Lesp of inﬂations of the the object (X, EX ) = F ((X, EX ),Y ). q ∗ In fact, each morphism (X, EX ) −→ (Z, EZ ) of Lesp) determines uniquely a morphism q ∗−1 ∞ ((X, EX ),Y ) −→ ((Z, EZ ), Z0), where CZ0 = q (CY ), which belongs to the class Lesp. This correspondence extends (uniquely) to a functor

Φ ∗ Y ∞ (X, EX )\Lesp −−−→ ((X, EX ),Y )\Lesp which is inverse to the functor

F! ∞ Y ∗ ((X, EX ),Y )\Lesp −−−→ (X, EX )\Lesp induced by F!. (b) By deﬁnition of the satellite, we have

!op S−G ◦ F (X, EX ) = S−G((X, EX ),Y ) cq q = lim Ker(G((Z, EZ ) −→ Cok((X, EX ) −→ (Z, EZ ))) ,

q where (X, EX ) −→ (Z, EZ ) runs through inﬂations of (X, EX ). On the other hand,

!op S−(G ◦ F )((X, EX ),Y ) = !op cq q lim Ker(G ◦ F ((Z, EZ ) −→ Cok(((X, EX ),Y ) −→ ((Z, EZ ), Z0)) = cq q lim Ker(G((Z, EZ ) −→ Cok((X, EX ) −→ (Z, EZ ))) ,

q where ((X, EX ),Y ) −→ ((Z, EZ ), Z0) runs through inflations of ((X, EX ),Y ). It follows from (a) above that these two limits are identical. 8.9.2. Constructions related to the functor F∗. Fix an object of the category r ∗ ℘ Esp∞ and an inflation (i.e. a morphism of Lesp)(Y, EY ) −→ (Z0, EZ0 ) with an inverse ℘∗ image functor CZ0 −→ CY . Let CZ(℘) denote the category whose objects are triples ∗ (M,L; e), where M ∈ ObCX ,L ∈ ObCZ0 , and e is a deflation M −→ ℘ (L). Morphisms ξ γ from (M,L; e) to (M 0,L0; e0) are given by pairs of arrows M −→ M 0,L −→ L0 such that the diagram e M −−−→ ℘∗(L)     ∗ ξ y y ℘ (γ) e M 0 −−−→e ℘∗(L0)

114 commutes. The composition is deﬁned obviously. p∗ π∗ There are natural functors (projections) CZ0 ←−−− CZ(℘) −−−→ CX deﬁned by

(ξ,γ) ξ π∗((M,L; e) −→ (M 0,L0; e0)) = (M −→ M 0) (ξ,γ) γ p∗((M,L; e) −→ (M 0,L0; e0)) = (L −→ L0)

p ∗ ∗ The functor p has a right adjoint, CZ0 −−−→ CZ(℘), which assigns to each object ∗ γ 0 ∗ L of the category CZ0 the object (℘ (L),L; id℘ (L)) and to each morphism L −→ L the morphism (℘∗(γ),γ) ∗ ∗ 0 0 (℘ (L),L; id℘∗(L)) −−−−−−−→ (℘ (L ),L ; id℘∗(L0)).

η ∗ The adjunction morphism IdCZ(℘) −→ p∗p assigns to each object (M,L; e) of the (e,id ) L ∗ category CZ(℘) the morphism (M,L; e) −−−→ (℘ (L),L; id℘∗(L)). The other adjunction morphism is the identity. The latter implies that the functor p∗ is fully faithful, or, equivalently, p∗ is a localization functor. It follows from the construction that the diagram

℘∗ CZ −−−→ CY  0    ∗ p∗ y y j π∗ CZ(℘) −−−→ CX commutes. (t,u) 0 0 0 8.9.2.1. Proposition. The class EZ(℘) of all arrows (M,L; e) −−−→ (M ,L ; e ) of the category CZ(℘) such that t ∈ EX and u ∈ EZ0 is a structure of a right exact category on CZ(℘).

Proof. Obviously, the class EZ(℘) is closed under compositions and contains all isomorphisms. It remains to show that it is stable by a base change and deﬁnes a (t,u) 0 0 0 subcanonical topology. Let (M ,L ; e ) −−−→ (M,L; e) be a morphism of EZ(℘) and (f,g) (M 00,L00; e00) −−−→ (M,L; e) an arbitrary morphism. Since t and u are deﬂations, there are cartesian squares

t0 u0 M −−−→ M 00 L −−−→ L00 f  e      fe y cart y f and ge y cart y g (1) t u M 0 −−−→ ML0 −−−→ L whose horizontal arrows are deﬂations.

115 ∗ Since ℘ is an ’exact’ functor (CZ0 , EZ0 ) −→ (CY , EY ), it maps the right cartesian square (1) to a cartesian square. So we have a commutative diagram

e3 e1 M −−−→ M −−−→ M 00 3 1     00 ep1y cart p1y cart y e e ℘∗(u0) e00 2 ∗ ∗ 00 00 M2 −−−→ ℘ (Le) −−−→ ℘ (L ) ←−−− M      ∗   ∗  ep2y cart ℘ (ge) y cart y ℘ (g) y f (2) e0 ℘∗(u) e M 0 −−−→ ℘∗(L0) −−−→ ℘∗(L) ←−−− M x x 0   e   e t M 0 −−−→ M with cartesian squares as indicated. It follows that all horizontal arrows and upper and p ◦e 1 3 ∗ lower vertical arrows of (2) are deﬂations. Therefore, the composition M3 −−−→ ℘ (Le) is a deﬂation. On the other hand, it is easy to see that the outer square of (2),

e1e3 M −−−→ M 3  1   ep2ep1 y y f t M 0 −−−→ M is cartesian. Therefore, since the left square (1) is cartesian. there exists a unique isomor- σ 0 phism Mf −→ M3 such that fe = ep1ep2σ and t = e1e3σ. We denote by e the composition p ◦e e ∗ 1 3 ∗ Mf −→ ℘ (Le) of the isomorphism σ and the deﬂation M3 −−−→ ℘ (Le). It follows from this argument that (M,f Le;e) is an object of the category CZ(℘) and

(f,eeg) (M,f Le; e) −−−→ (M 0,L0; e0) e  0 0   (3) (t , u ) y y (t, u) (f,g) (M 00,L00; e00) −−−→ (M,L; e) is a cartesian square in CZ(℘) whose vertical arrows belong EZ(℘). Thus, EZ(℘) is (the class of covers of) a pretopology. This pretopology is subcanonical; i.e. for any (t, u) ∈ EZ(℘),, the corresponding cartesian square

(et,eu) (M,f Le; e) −−−→ (M 0,L0; e0) e  0 0   (t , u ) y y (t, u) (t,u) (M 0,L0; e0) −−−→ (M,L; e)

116 is cocartesian, because the squares

t0 u0 M −−−→ M 0 L −−−→ L0 f  e      et y cart y t and eu y cart y u t u M 0 −−−→ ML0 −−−→ L are cocartesian.

8.9.2.2. Lemma. Suppose that any pair N ←− M −→ L of morphisms of EX can be completed to a commutative square M −−−→ L     y y N −−−→ L whose arrows are belong to EX . Then the pair ((CZ(℘), EZ(℘)),CZ0 ), where CZ0 is iden- r tiﬁed with its image in CZ(℘), is an object of the category Esp∞ of right inﬁnitesimal ’spaces’.

Proof. Let (M,L; e) be any object of CZ(℘), and let

tn tn−1 t1 M = Mn −−−→ Mn−1 −−−→ ... −−−→ M0 be a CY -coﬁltration of the object M. By hypothesis, there exists a commutative square

tn M −−−→ M  n−1   0 e y y e (4) β0 ℘∗(L) −−−→ L0

0 ∗ whose all arrows are deflations. Since β is a deflation ℘ (L) ∈ ObCY , and CY is a ∗ 0 ℘ topologizing subcategory, of (CX , EX ), L ∈ ObCY . Since CZ0 −→ CY is a localization 0 ∗ 00 00 functor, the object L is isomorphic to an object ℘ (L ) for some L ∈ ObCZ0 . The ℘ ∗ morphism (Y, EY ) −−−→ (Z0, EZ0 ) belongs to the class Lesp. Therefore, there exists an ∗ 00 ∼ ∗ γn isomorphism ℘ (L ) −→ ℘ (Ln−1) and a deflations L −→ Ln−1 such that the diagram

℘∗(γ ) ∗ n ∗ ℘ (L) −−−→ ℘ (Ln−1)  x 0  β y o ∼ L0 −−−→ ℘∗(L00) commutes. Combining this with the diagram (4), we obtain the commutative diagram

tn M −−−→ M  n−1   e y y en−1 ℘∗(γ ) ∗ n ∗ ℘ (L) −−−→ ℘ (Ln−1)

117 whose all arrows are deﬂations. Continuing this process, we obtain a commutative diagram

tn tn−1 t1 M −−−→ M −−−→ ... −−−→ M  n−1 0    e y y en−1 y e0 ℘∗(γ ) ℘∗(γ ) ℘∗(γ ) ∗ n ∗ n 1 ∗ ℘ (L) −−−→ ℘ (Ln−1) −−−→ ... −−−→ ℘ (L0)

which encodes a CZ0 -coﬁltration

(tn,γn) (tn−1,γn−1) (t1,γ1) (M,L; e) −−−−−−−→ (Mn−1,Ln−1; en−1) −−−−−−−→ ... −−−−−−−→ (M0,L0; e0) of the object (M,L; e) in the right exact category (CZ(℘), EZ(℘)). r 8.9.3. Proposition. Let ((X, EX ),Y ) be an object of Esp∞. Suppose that the right exact category (CX , EX ) satisﬁes the following conditions: e f (a) Any pair of arrows L ←− M −→ M1 of CX , where e is a deﬂation, can be completed to a commutative square

f M −−−→ M  1   0 e y y e (3) fe L −−−→ L1 where e0 is a deflation too. (b) If both morphisms e and f in the condition (a) are deflations, then the morphisms e0 and fe in the diagram (3) can be chosen to be deflations. Then the natural morphism

∗op ∗op S−G(Y, EY ) = S−G ◦ F ((X, EX ),Y ) −−−→ S−(G ◦ F )((X, EX ),Y ) (4)

r ∞ op is an isomorphism for any functor G from (Esp∞, Lesp) to any category CZ with initial objects and limits of ﬁltered systems. ∗ r ∞ Here the functor F is viewed as a morphism from the left exact category (Esp∞, Lesp) ∗ ∗ to the left exact category (Espr , Lesp).

Proof. (i) Let CX denote the category whose objects are triples (M, L; e), where M ∈ 0 0 0 ObCX , L ∈ ObCY , and e is a deﬂation M −→ L. Morphisms (M, L; e) −→ (M , L ; e ) f g are pairs of arrows M −→ M0, L −→ L0 such that the diagram

e M −−−→ L     f y y g e0 M0 −−−→ L0

118 commutes. The composition is defined naturally. (t,u) 0 0 0 (ii) The class EX of all morphisms of CX (M, L; e) −−−→ (M , L ; e ) such that both t 0 u 0 M −→ M and L −→ L are deflations, is a right exact structure on the category CX. This fact is a special case of 8.9.2.1. ρ∗ (iii) The functor CX −→ CY which maps each object (M,L; e) of CX to L and each morphism (f, g) to g is a continuous localization with the canonical right adjoint which maps each object L of CY to the object (L, L; idL). ∗ ℘ (iv) Fix an inflation (– a morphism of Lesp)(Y, EY ) −→ (Z0, EZ0 ) with an inverse ℘∗ q∗ image functor CZ0 −→ CY . Let CZ(℘) −→ CX be the functor which assigns to each ∗ object (M.L; e) of CZ(℘) the object (M, ℘ (L); e) of the category CX and to each morphism (f,g) (f,℘∗(g)) (M, L; e) −−−→ (M0, L0; e0) the morphism (M, ℘∗(L); e) −−−→ (M0, ℘∗(L0); e0). It is easy to see that the square

q∗ CZ −−−→ C  ℘ X ∗  ∗ p y y ρ ℘∗

CZ0 −−−→ CY is cartesian. In particular, since ℘∗ and ρ∗ are ’exact’ localizations, the functors p∗ and q∗ are ’exact’ localizations too. π∗ (v) The functor CX −→ CX , (M,L; e) 7→ M, is an ’exact’ localization functor ∗ −1 eπ (CX, EX) −→ (CX , EX ). In other words, the unique functor Σq∗ CX −→ CX is an equivalence of categories.

(v’) It follows from the deﬁnition of right exact structure on CX and (the argument of) 8.9.2.1 that the functor π∗ is ’exact’. (v”) It is easy to verify that the conditions (a) and (b) imply that Σq∗ is a left multiplicative system.

(v”’) For each M ∈ ObCX , we choose an object (M,LM ; tM ) of the category CX and set πe∗(M) = (M,LM ; tM ), where (M,LM ; tM ) is regarded as an object of the quotient −1 f category Σq∗ CX. Thanks to the condition (b), for any morphism M −→ N, there exists a commutative square f M −−−→ N     tM y y e fe LM −−−→ Ne with e ∈ EX . By the condition (b), there exists a commutative square

tN N −−−→ L  N   e y y e t0 Ne −−−→ L

119 whose all arrows are deﬂations. Thus, we have morphisms

(f,t0◦f) (id ,e) e 0 N e (M,LM ; tM ) −−−−−−−→ (N, L; t e) ←−−−−−−− (N,LN ; tN )

Since the left arrow here belongs to Σq∗ , this pair of morphisms determines a morphism π (f) e∗ −1 (M,L ; t ) = π (M) −−−→ π (N) = (N,L ; t ) of Σ ∗ C . The map π is a functor M M e∗ e∗ N N q X e∗ −1 ∗ C −→ Σ ∗ C . It follows from the construction that π π = Id . On the other hand, it X q X e e∗ CX follows from the condition (b) that, for each (M,L; e) ∈ ObCX, there exists an isomorphism ∗ (M,L; e) −→ πe∗πe (M,L; e) = (M,LM ; tM ). Altogether shows that πe∗ is a quasi-inverse to ∗ the functor πe . (vi) The constructions of (iv) and (v) assigns, in a functorial way, to each morphism ∗ ℘ ∗ ℘ (Y, EY ) −→ (Z0, EZ0 ) of Lesp) with an inverse image functor CZ0 −→ CY two ’exact’ ∗ ∗ q π ∗ ∗ localizations, CZ(℘) −→ CX and CX −→ CX . Their composition, π q , is an inverse q℘ image functor of an inflation ((X, EX ),Y ) −−−→ ((Z℘, EZ℘ ), Z0) of infinitesimal right ℘ exact ’spaces’ which ’lifts’ the deflation (Y, EY ) −−−→ (Z0, EZ0 ) of right exact ’spaces’. (vii) By the argument of 3.3.2 and the observation 8.8.3.1, we have

∗op ∗op 0 0 cq 00 00 S−(G ◦ F )((X, EX ),Y ) = lim Ker G ◦ F (((X , EX0 ),Y ) −→ ((X , EX00 ),Y ) (5) 0 c℘ 00 = lim Ker G((Y , EY 0 ) −→ (Y , EY 00 ) ,

q 0 0 ∞ where ((X, EX ),Y ) −−−→ ((X , EX0 ),Y ) runs through the category ((X, EX ),Y )\Lesp ℘ 0 of inflations of ((X, EX ),Y ) and (Y, EY ) −→ (Y , EY 0 ) is the inflation of (Y, EY ) deter- q℘ mined by the inflation q. Since the inflation ((X, EX ),Y ) −−−→ ((Z℘, EZ℘ ), Z0) (see ℘ 0 (vi) above) induces the same inflation (Y, EY ) −−−→ (Y , EY 0 ), the arbitrary inflations of ((X, EX ),Y ) can be replaced by the inflations of the form q℘, where ℘ runs through r the category (Y, EY )\Lesp of inflations of the right exact category (Y, EY ). But, the limit 0 c℘ 00 r of Ker G((Y , EY 0 ) −→ (Y , EY 00 ) , where ℘ runs through the category (Y, EY )\Lesp is isomorphic to S−G(Y, EY ). 8.9.4. The subcategory of ’spaces’ Espo. We denote this way the full subcategory ∗ of the category Espr whose objects are right exact ’spaces’ (X, EX ) such that CX has final objects and all morphisms to final objects are deflations. Thus, we have a left exact o o category (Espo, Lesp), where Lesp is the restriction to the subcategory Espo of the left r ∗ exact structure Lesp on Espr . We denote by Esp∞ the full subcategory of the category Esp∞ of right exact infinites- o r imal ’spaces’ generated by all ((X, EX ),Y ) such that (X, EX ) is an object of Espo. We denote by L∞ the restriction of the left exact structure L∞ to the subcategory Esp∞. o esp o ∗ ! The functors F∗, F , and F (cf. 8.8) induce the functors

! ∗ F∗ F F o e o o e o o e o Esp −−−→ Esp∞, Esp∞ −−−→ Esp , and Esp∞ −−−→ Esp .

120 8.9.5. Proposition. Let CZ be a category with initial objects and limits of ﬁltered o op systems; and let G be a functor (Espo, Lesp) −−−→ CZ . Then the natural morphisms

!op !op ∗op ∗op S−G ◦ Fe −−−→ S−(G ◦ Fe ) and S−G ◦ Fe −−−→ S−(G ◦ Fe ) are isomorphisms. Here the functors Fe! and Fe∗ are viewed as morphisms from the left exact category (Esp∞, L∞) to the left exact category (Espo, Lo ). o o esp o Proof. Fix an object ((X, EX ),Y ) of the category Esp∞. (a) By the argument (a) of 8.9.1, the functor Fe! induces an isomorphism from the category ((X, E ),Y )\L∞ of inﬂations of ((X, E ),Y ) onto the category (X, E )\Lo of X o X X esp ! inﬂations of the object (X, EX ) = Fe ((X, EX ),Y ), which implies a canonical isomorphism

!op !op S−G(X, EX ) = S−G ◦ Fe ((X, EX ),Y ) −−−→ S−(G ◦ Fe )((X, EX ),Y ).

f (b) If ((X, E ),Y ) is an object of Esp∞, than any pair of arrows L ←−e M −→ N of X o CX is a part of the commutative square

f M −−−→ N     e y y tN tL L −−−→ x• where x• is a ﬁnal object of CX , hence, by hypothesis, the arrows tL and tN are deﬂations. Therefore, the conditions of 8.9.3 hold, which implies that the canonical morphism

∗op ∗op S−G(Y, EY ) = S−G ◦ F ((X, EX ),Y ) −−−→ S−(G ◦ F )((X, EX ),Y )

r ∞ op is an isomorphism for any functor G from (Esp∞, Lesp) to any category CZ with initial objects and limits of ﬁltered systems.

8.9.6. Corollary. Let (CZ , EZ ) be a right exact category with initial objects and limits ∗ o op of ﬁltered systems. Then for any universal ∂ -functor G• from (Espo, Lesp) to (CZ , EZ ), op op the compositions G ◦ F! and G ◦ F∗ are universal ∂∗-functors from (Esp∞, L∞)op to • e • e o o ∗ ∗ !op ∗op (CZ , EZ ). If the ∂ -functor G• is ’exact’, then the ∂ -functors G• ◦ Fe and G• ◦ Fe are ’exact’. Proof. The assertion follows from 8.9.4, 5.3.2, and the fact that Fe! and Fe∗ are ’exact’ functors from the left exact category (Esp∞, L∞) to the left exact category (Espo, Lo ). o o esp Details are left to the reader.

∗ o op 8.9.7. Corollary. Let G• be a universal ∂ -functor from (Espo, Lesp) to a category ρop ∗op F !op CY . Suppose that the functor G maps a natural morphism Fe −→ Fe to an isomorphism. op G•ρ ∗op F !op ∗ Then G• ◦Fe −−−→ G• ◦Fe is an isomorphism of ∂ -functors. In other words, for any

121 o ∼ object ((X, EX ),Y ) of the category Esp∞, there is a natural isomorphism G•(Y, EY ) −→ G•(X, EX ). ∗ ∗op !op Proof. By 8.9.7, the ∂ -functors G• ◦ Fe and G• ◦ Fe are universal. Therefore ∗op !op a morphism from G• ◦ Fe to G• ◦ Fe is an isomorphism iﬀ its zero component is an isomorphism, whence the assertion.

o o o 8.9.8. Corollary. Let K• = (Ki , di | i ≥ 0) be the K-functor on the left exact o × o ∗ category (Esp , Lesp) of right exact ’spaces’; that is K• is a universal ∂ -functor from op o o op o ! (Esp , Lesp) whose zero component is (X, EX ) 7−→ K0(X, EX ). Then K• ◦ Fe and op Ko ◦ F∗ are ’exact’, universal ∂∗-functors from (Esp∞, L∞)op to − mod. • e o o Z Proof. This is a special case of 8.9.6. r 8.10. The k-linear setting. Instead of the left exact category (Espr, Lesp) of right r r exact ’spaces’, we consider the left exact category (Espk, Lek) of right exact k-linear ’spaces’ r r r −1 r (cf. Q8.6). Here Lek is the left exact structure induced by Lesp; i.e. Lek = Fr (Lesp), where r Fr is the natural forgetful functor Espk −→ Espr. 8.10.1. The left exact category of right exact infinitesimal k-’spaces’. A right exact infinitesimal k-’space’ is a pair ((X, EX ),Y ), where (CX , EX ) is a right exact k-linear category and CY a topologizing subcategory of (CX , EX ) such that CX = (CY )(∞). f A morphism from ((X, EX ),Y ) to ((X, EX), Y) is a morphism (X, EX ) −→ (X, EX) right ∗ exact k-’spaces’ which maps Y to Y; i.e. f is a k-linear ’exact’ functor from (CX, EX) to ∗ ∞ (CX , EX ) such that f (CY) ⊆ CY . This defines a category denoted by Espr,k. The left ∞ r ∞ r exact structure Lesp on Esp∞ induces (via the forgetful functor Espr,k−−−→Esp∞, a left ∞ ∞ exact structure, Lr,k, on Espr,k. r ∞ We denote by Fk∗ the embedding Espk −→ Espr,k which assigns to each object (X, EX ) ∞ the object ((X, EX ),X) of the category Espr,k. This functor is fully faithful and has a left adjoint F∗ ∞ k r Espr,k −−−→ Espk, ((X, EX ),Y ) 7−→ (Y, EY ), and a right adjoint

F! ∞ k r Espr,k −−−→ Espk, ((X, EX ),Y ) 7−→ (X, EX ).

∗ ! All three functors, Fk∗, Fk, and Fk, are ’exact’.

8.10.2. Proposition. Let CZ be a k-linear category with limits of ﬁltered systems; r r op and let G be a functor (Espk, Lek) −−−→ CZ . Then the natural morphisms

!op !op ∗op ∗op S−G ◦ Fk −−−→ S−(G ◦ Fk ) and S−G ◦ Fk −−−→ S−(G ◦ Fk ) are isomorphisms. Here the functors F! and F∗ are viewed as morphisms from the left ∞ ∞ r r exact category (Espr,k, Lr,k) to the left exact category (Espk, Lek). ∞ Proof. Fix an object ((X, EX ),Y ) of the category Espr,k.

122 (a) By the (k-linear version of the) argument (a) of 9.9.1, the functor F! induces an ∞ isomorphism from the category ((X, EX ),Y )\Lesp of inﬂations of ((X, EX ),Y ) onto the ∗ ! category (X, EX )\Lesp of inﬂations of the the object (X, EX ) = F ((X, EX ),Y ), which implies a canonical isomorphism

!op !op S−G(X, EX ) = S−G ◦ Fk ((X, EX ),Y ) −−−→ S−(G ◦ Fk )((X, EX ),Y ).

(b) Similarly, the k-linear version of the arguments of 8.9.5 and 8.9.3 shows that the ∗op ∗op canonical morphism S−G ◦ Fk −−−→ S−(G ◦ Fk ) is an isomorphism.

8.10.3. Corollary. Let (CZ , EZ ) be a k-linear right exact category with limits of ∗ r r op ﬁltered systems. Then for any universal ∂ -functor G• from (Espk, Lek) to (CZ , EZ ), !op ∗op ∗ ∞ ∞ op the compositions G• ◦ Fk and G• ◦ Fk are universal ∂ -functors from (Espr,k, Lr,k) to ∗ ∗ !op ∗op (CZ , EZ ). If the ∂ -functor G• is ’exact’, then the ∂ -functors G• ◦ Fk and G• ◦ Fk are ’exact’. Proof. See the argument of 8.9.6.

r r r 8.10.4. Corollary. Let K• = (Ki , di | i ≥ 0) be the K-functor on the left exact r r r ∗ category (Espk, Lek) of right exact k-linear ’spaces’; that is K• is a universal ∂ -functor r r op r !op from (Espk, Lek) whose zero component is (X, EX ) 7−→ K0(X, EX ). Then K• ◦ Fk and r ∗op ∗ ∞ ∞ op K• ◦ Fk are ’exact’, universal ∂ -functors from (Espr,k, Lr,k) to Z − mod. Proof. This is a special case of 8.10.3. 8.11. An application to K-functors: devissage.

8.11.1. Proposition. (Devissage for K0.) Let ((X, EX ),Y ) be an inﬁnitesimal ’space’ such that (X, EX ) has the following property (which appeared in 8.4.1): t1 t2 (‡) for any pair of deﬂations M1 ←− M −→ M2, there is a commutative square

t1 M −−−→ M  1   t2 y y p2 p1 M2 −−−→ M3

of deﬂations such that the unique morphism M −→ M1 ×M3 M2 is a deﬂation. Then the natural morphism

K0(Y, EY ) −−−→ K0(X, EX ) (1) is an isomorphism.

Proof. Let M be an object of CX . Since CX = (CY )(∞), there exists a CY -coﬁltration

en e2 e1 M = Mn −−−→ ... −−−→ M1 −−−→ M0. (2)

123 That is the arrows of (2) are deflations such that the object M0 and objects Ker(ei) belong to the subcategory CY for every 1 ≤ i ≤ n. Since the subcategory CY is (semi)topological, any refinement of a CY -cofiltration is a CY -cofiltration. (a1) The map X [M] 7−→ [M0]CY + [Ker(ei)]CY (3) 1≤i≤n applied to a refinement of the cofiltration (2) gives the same result. Here [N]CY denotes the image of the object N in K0(Y ). tm t2 t1 In fact, for any sequence of deflations Mm −→ ... −→ M1 −→ M0, we have a commutative diagram

tm tm−1 t3 t2 t1 K −−−→e K −−−→e ... −−−→e K −−−→e K −−−→e x m m−1 2 1       (4) y cart y cart . . . cart y cart y cart y tm tm−1 t3 t2 t1 Mm −−−→ Mm−1 −−−→ ... −−−→ M2 −−−→ M1 −−−→ M0 formed by cartesian squares. Here x is an initial object of the category CX . Since the ’composition’ of cartesian squares is a cartesian square, it follows that K1 = Ker(t1), K2 = Ker(t1t2),..., Km = Ker(t1t2 ... tm). Since each square

t` K −−−→e K ` `−1   y cart y t` M` −−−→ M`−1 of the diagram (4) is cartesian, all morphisms et` are deflations and Ker(et`) ' Ker(t`) for all 1 ≤ ` ≤ m. Therefore, X X [Ker(t1t2 ... tm)] = [Km] = [Ker(eti)] = [Ker(ti)]. (5) 1≤i≤n 1≤i≤n This shows that the right hand side of (3) remains the same when each of the deflations ei is futher decomposed into a sequence of deflations. (a2) By 8.4.1, any two finite cofiltrations of an object of CX have equivalent refinements. Together with (a1) above, this implies that the map (3) does not depend on the choice of CY -cofiltrations of objects. Thus, (3) defines a map, ψe, from the set |CX | of ∗ isomorphism classes of objects of the category CX to the group K0 (Y, EY ). 0 j e 00 (a3) For any conflation M −→ M −→ M in (CX , EX ), we have

ψe([M]) = ψe([M 0]) + ψe([M 00]).

Indeed, let M −→ ... −→ M0 be some CY -cofiltration of M. By 8.4.1, this cofiltration and the cofiltration M −→e M 00 have equivalent refinements which are, forcibly, CY -cofiltrations. Consider the obtained this way refinement e e e e e n m 00 m 2 1 M = Mn −−−→ ... −−−→ Mm−1 = M −−−→ ... −−−→ M1 −−−→ M0

124 and the associated commutative diagram

en en−1 em K −−−→e K −−−→e ... −−−→e x n n−1     (6) y cart y cart . . . cart y e en−1 e em−1 e n m 00 1 M −−−→ Mn−1 −−−→ ... −−−→ M −−−→ ... −−−→ M0

e 00 built of cartesian squares. Here x is an initial object of the category CX . Since M −→ M equals to the composition em ◦ ... ◦ en, it follows from the argument of (a1) that Kn ' 0 0 M = Ker(e) and the upper row of (6) is a CY -ﬁltration of the object M . The latter implies that X X ψ([M 0]) = [Ker(e )] = [Ker(e )] e ei CY i CY m≤i≤n m≤i≤n (see (5) above). From the lower row, we obtain

00 X ψe([M ]) = [M0]CY + [Ker(ei)]CY and m

Therefore, ψe([M]) = ψe([M 0]) + ψe([M 00]). ψe (a4) The map |CX | −→ K0(Y, EY ) extends uniquely to a Z-module morphism

Zψe Z|CX | −−−→ K0(Y, EY ). (7)

It follows from (a3) that the morphism (7) factors through a (uniquely determined) Z- module morphism ψ0 K0(X, EX ) −−−→ K0(Y, EY ).

The claim is that the morphism ψ0 is invertible and its inverse is

K0(j) K0(Y, EY ) −−−→ K0(X, EX ).

It is immediate that ψ0 ◦ K0(j) = id . K0(Y,EY ) The equality K0(j) ◦ ψ0 = id is also easy to see.: if M is an object of CX K0(X,EX ) en e2 e1 endowed with a CY -coﬁltration M = Mn −→ ... −→ M1 −→ M0, then X X K0(j) ◦ ψ0([M]) = K0(j)([M0]CY + [Ker(ei)]CY ) = [M0] + [Ker(ei)] = [M]. 1≤i≤n 1≤i≤n

This proves the assertion.

125 8.11.2. The ∂∗-functor Ksq. Let Les denote the left exact structure on the category • o o es Esp of Espr (cf. 8.9.4) induced by the (deﬁned in 6.8.3.3) left exact structure Lsq on the sq category Espr of right exact ’spaces’. Let Ki (X, EX ) denote the i-th satellite of the functor K with respect to the left exact structure Les. 0 o

8.11.3. Proposition. Let ((X, EX ),Y ) be an infinitesimal ’space’ such that the right exact ’space’ (X, EX ) has the property (‡) of 9.11.1, the category CX has final objects, and all morphisms to final objects are deflations. Then the natural morphism

sq sq Ki (Y, EY ) −−−→ Ki (X, EX ) (8) is an isomorphism for all i ≥ 0. ∞ ∞ Proof. (a) Consider the full subcategory Espr,‡ of the category Espr of infinitesimal ’spaces’ whose objects are infinitesimal ’spaces’ ((X, EX ),Y ) such that (X, EX ) satisfies ∞ es∞ the property (‡). The claim is that for any functor G from (Espr , Lsq ) to a category CZ with filtered limits, the satellite of the composition of G with the inclusion functor ∞ ∞ Espr,‡ −→ Espr is naturally isomorphic to the composition of the satellite of G with the inclusion functor. In fact, it is easy to see that if the right exact ’space’ (X, EX ) has the property (‡), then this property holds for the right exact ’space’ Pa(X, EX ) of paths of (X, EX ). By es 6.11.1, Pa(X, EX ) is an injective object of the left exact category (Espr, Lsq) and the canonical morphism (X, EX ) −→ Pa(X, EX ) is an inflation. (b) One can check that 8.9.5 (hence its corollary 8.9.6) remains true, when the left o o es exact structure Lesp is replaced by the left exact structure on Esp induced by Lsq. (c) The assertion follows now from (a) above, 8.11.1, and 8.9.6.

126 Complementary facts. C1. Complements on kernels and cokernels. C1.1. Kernels of morphisms of ’spaces’. The category |Cat|o of ’spaces’ has an initial object x represented by the category with one object and one (identical) morphism. By [KR, 2.2], the category |Cat|o has small limits (and colimits). In particular, any f morphism of |Cat|o has a kernel. The kernel of a morphism X −→ Y of |Cat|o can be explicitly described as follows. f ∗ Let CY −→ CX be an inverse image functor of f. For any two objects L, M of the category CX , we denote by If (L, M) the set of all arrows L −→ M which factor ∗ through an object of the subcategory f (CY ). The class If of arrows of CX obtained this way is a two-sided ideal; i.e. it is closed under compositions on both sides with arbitrary arrows of CX . We denote by CXf the quotient of the category CX by the ideal If ; that is ObCXf = ObCX , CXf (L, M) = CX (L, M)/If (L, M) for all objects L, M, and the compostion is induced by the composition in CX . Each object M of the image of the ∗ subcategory f (CY ) in CXf has the property that CXf (L, M) and CXf (M,L) consist of at ∗ most one arrow. This allows to deﬁne a category CK(f) by replacing the image of f (CY ) ∗ by one object z and one morphism, idz. (i.e. ObCK(f) = ObCX /f (CY )). If objects L and

M are not equal to z, then we set CK(f)(L, M) = CXf (L, M). The set CK(f)(L, z) (resp. CK(f)(z, M)) consists of one element iﬀ there exists a morphism from L to an object of ∗ ∗ f (CY ) (resp. from an object of f (CY ) to M); otherwise, it is empty. ∗ We denote by k(f) the natural projection CX −→ CK(f). Thus, we have a commutative square of functors c(f)∗ CK(f) ←−−− CX x x ∗   ∗ πz   f Cx ←−−− CY ∗ where πz maps the unique object of Cx to z. This square corresponds to a cartesian square

c(f) K(f) −−−→ X     πz y y f x −−−→ Y

c(f) f of morphisms of ’spaces’; i.e. the morphism K(f) −−−→ X is the kernel of X −→ Y . Similarly to Sets, the category |Cat|o has a unique final object represented by the empty category. Since there are no functors from non-empy categories to the empty category, the cokernel of any morphism of |Cat|o is the unique morphism to the final object. C1.2. Kernels and cokernels of morphisms of relative objects. Fix an object V of a category CX and consider the category CX /V . This category has a final object, (V, idV ), so we can discuss cokernels of its morphisms. Notice that the forgetful functor CX /V −→ CX is exact, in particular, it preserves push-forwards. Therefore, the kernel

127 f a a of a morphism (M, g) −→ (N, h) exists iﬀ a push-forward N V = N V exists and is M f,g a a h0 h equal to (N V, h0), where N V −→ V is determined by N −→ V . M M

C1.2.1. Kernels. If the category CX has an initial object x, then (x, x → V ) is an initial object of the category CX /V . The forgetful functor CX /V −→ CX preserves pull- backs; in particular, it preserves kernels of morphisms. So that the kernel of a morphism f k(f) f (M, g) −→ (N, h) exists iﬀ the kernel Ker(f) −−−→ M of M −→ N exists; and it is equal k(f) to (Ker(f), g ◦ k(f)) −−−→ (M, g). C1.3. Application: cokernels of morphisms of relative ’spaces’. Fix a ’space’ S and consider the category |Cat|o/S of ’spaces’ over S. According to C1.2, the cokernel f of a morphism (X, g) −→ (Y, h) of ’spaces’ over S is the pair (Cok(f), eh), where CCok(f) is the pull-back (in the pseudo-categorical sense) of the pair of inverse image functors g∗ h∗ CS −→ CX ←− CY . That is objects of the category CCok(f) are triples (M,N; φ), where ∗ ∼ ∗ M ∈ ObCS ,N ∈ ObCY and φ an isomorphism g (M) −→ f (N). Morphisms from (M,N; φ) to (M 0,N 0; φ0) are given by a pair of arrows M −→u M 0, N −→v N 0 such that the square g∗(u) g∗(M) −−−→ g∗(M 0)     0 φ yo oy φ f ∗(v) f ∗(N) −−−→ f ∗(N 0)

eh∗ commutes. The functor CS −−−→ CCok(f) which assigns to every object L of the category ∗ ∗ ∗ ∼ ∗ ∗ CS the object (g (L), h (L); ψ(L)), where ψ is an isomorphism g −→ f h , is an inverse image functor of the morphism eh. C1.4. Categories with initial objects and associated pointed categories. Let

CX be a category with an initial object, x. Then the category CXx = CX /x is a pointed category with a zero object (x, idx).

C1.4.1. Example: augmented monads. Let CX be the category Mon(X) of monads on the category CX . The category CX has a canonical initial object x which is + the identical monad (IdCX , id). The category CXx coincides with the category Mon (X) of augmented monads. Its objects are pairs (F, ), where F = (F, µ) is a monad on CX and is a monad morphism F −→ (IdCX , id) called an augmentation morphism. One can see that a functor morphism F −→ IdCX is an augmentation morphism iﬀ (M, (M)) is an F-module morphism for every M ∈ ObCX . In other words, there is a bijective correspondence between augmentation morphisms and sections CX −−−→ F − mod of the f∗ forgetful functor F − mod −−−→ CX . C1.5. Pointed category of ’spaces’. Consider ﬁrst a simpler case – the category Catop. It has an initial object, x, which is represented by the category with one object and

128 one (identical) morphism. The associated pointed category Catop/x is equivalent to the category whose objects are pairs (X, OX ), where X is a ’space’ and OX an object of the ∗ category CX representing X. Morphisms from (X, OX ) to (Y, OY ) are pairs (f , φ), where ∗ ∗ f is a functor CY −→ CX and φ is an isomorphism f (OY ) −→ OX . The composition of (f ∗,φ) (g∗,ψ) ∗ ∗ ∗ ∗ ∗ (X, OX ) −−−→ (Y, OY ) −−−→ (Z, OZ ) is given by (g , ψ) ◦ (f , φ) = (f ◦ g , φ ◦ f (ψ)). The pointed category |Cat|o/x associated with the category of ’spaces’ |Cat|o admits a similar realization after ﬁxing a pseudo-functor

c o op ∗ ∗ f,g ∗ ∗ |Cat| −−−→ Cat ,X 7−→ CX , f 7−→ f ;(gf) −−−→ f g ,

– a section of the natural projection Catop −→ |Cat|o. Namely, it is equivalent to a category o |Cat|x whose objects are (as above) pairs (X, OX ), where OX ∈ ObCX , morphisms from (X, OX ) to (Y, OY ) are pairs (f, φ), where f is a morphism of ’spaces’ X −→ Y and φ is an (f,φ) (g,ψ) ∗ isomorphism f (OY ) −→ OX . The composition of (X, OX ) −−−→ (Y, OY ) −−−→ (Z, OZ ) ∗ is the morphism (g ◦ f, φ ◦ f (ψ) ◦ cf,g). C1.5.1. Cokernels of morphisims. One can deduce from the description of coker- o nels in C1.3 in terms of the realization of the category |Cat|x given above, that the cokernel (f,φ) (c(f),ψ) of a morphism (X, OX ) −−−→ (Y, OY ) is isomorphic to (Y, OY ) −−−→ (C(f), OC(f)), where ∗ CC(f) is a subcategory of CY whose objects are M ∈ ObCY such that f (M) 'OY and morphisms are all arrows between these objects which f ∗ transforms into isomorphisms. The ’structure’ object OC(f) coincides with OY ; the inverse image functor of c(f) is the inclusion functor CC(f) −→ CY ; and the isomorphism ψ is identical. C1.6. The category of k-’spaces’. We call ’spaces’ represented by k-linear additive o categories k-spaces. We denote by |Catk| the category whose objects are k-’spaces’ and morphisms X −→ Y are isomorphism classes of k-linear functors CY −→ CX . The category o |Catk| is pointed: its zero object is represented by the zero category. It is easy to see that f c every morphism X −→ Y has a canonical cokernel Y −→ Cok(f), where CCok(f) is the ∗ subcategory Ker(f ) of CY (– the full subcategory generated by all objects L such that ∗ ∗ ∗ f (L) = 0) and c is the inclusion functor Ker(f ) −→ CY . k(f) The kernel Ker(f) −−−→ X of f admits a simple description which is a linear version of the one in C1.1. Namely, CKer(f) is the quotient of the category CX by the ideal If ∗ formed by all morphisms of CX which factor through objects of f (CY ). The inverse image ∗ of k(f) is the canonical projection CX −→ CX /If . o C1.6.1. k-’Spaces’ over Sp(k). Consider now the full subcategory |Catk|Sp(k) of the category of k-’spaces’ over the aﬃne scheme Sp(k) whose objects are pairs (X, f) f ∗ where X −→ Sp(k) is continuous (i.e. f has a right adjoint, f∗). This category admits a realization in the style of C1.5. Namely, it is equvalent to the category whose objects are pairs (X, OX ), where X is a k-’space’ and OX is an object of the category CX such that there exist inﬁnite coproducts of copies of OX and cokernels of morphisms between ∗ these coproducts. Morphisms from (X, OX ) to (Y, OY ) are pairs (f, φ), where f is a

129 ∗ k-linear functor CY −→ CX and φ an isomorphism f (OY ) −→ OX . The composition is deﬁned as in C1.5 (see [KR, 4.5]). By C1.2, kernels of morphisms (as well as other limits) (f,φ) o are inherited from |Catk| . That is the kernel of a morphism (X, OX ) −−−→ (Y, OY ) is (k(f),id) ∗ the morphism (Ker(f), OKer(f)) −−−→ (X, OX ), where CKer(f) = CX /If , k(f) is the canonical projection CX −→ CX /If , and OKer(f) is the image of OX . (c(f),ψ)

The cokernel (Y, OY ) −−−→ (Cf , OCf ) of (f, φ) is described following C1.3. Objects of the category CCf are triples (M,N; φ), where M ∈ ObCY ,N ∈ Obk − mod, and α is an ∗ ∗ ∗ isomorphism f (M) −→ γ (N). Here γ is a functor k − mod −→ CX which maps k to ∗ OX and preserves colimits (which determines γ uniquely up to isomorphism). Morphisms are deﬁned as in C1.3. The structure object OCf is (OY , k, φ). The inverse image functor c(f)∗ of c(f) is the projection (M,N; α) 7−→ M.

C1.7. The (bi)categories Cat? and Catpt. Let Cat? denote the category whose objects are pairs (CX , x), where CX is a category and x its initial object; morphisms (CX , x) −→ (CY , y) are pairs (F, φ), where F is a functor CX −→ CY and φ a morphism (F,φ) (G,γ) F (x) −→ y. The composition of two morphisms, CX , x) −−−→ (CY , y) −−−→ (CZ , z), is given by (G, γ) ◦ (F, φ) = (G ◦ F, γ ◦ G(φ)). (F,φ) Fφ Every morphism CX , x) −−−→ (CY , y) deﬁnes a functor CXx −→ CYy between the corresponding pointed categories, and the map (F, φ) 7−→ Fφ respects compositions and maps identical morphisms to identical functors; i.e. the correspondence

(CX , x) 7−→ CXx , (F, φ) 7−→ Fφ is a functor, J∗, from the category Cat? onto the full subcategory Catpt of Cat whose ob- J∗ jects are pointed categories. The functor J∗ is a right adjoint to the functor Catpt −→ Cat? which assigns to each pointed category CX an object (CX , x) of the category Cat? and to F (F,φ) every functor CX −→ CY between pointed categories the morphism CX , x) −−−→ (CY , y) φ η ∗ in which the arrow F (x) −→ y is uniquely deﬁned. The adjunction arrow IdCatpt −→ J∗J ∼ assigns to each pointed category CX the natural isomorphism CX −→ CXx (where x is ∗ the zero object of CX involved in the deﬁnition of J ). The other adjunction arrow, ∗ J J∗ −→ IdCat? , assigns to each object (CY , y) of Cat? the forgetful functor CYy −→ CY . Notice by passing that the image of this forgetful functor is the full subcategory of CY generated by all objects having a morphism to an initial object.

C1.8. Induced right exact structures. A pretopology τ on CX induces a pretopology τV on the category CX /V for any V ∈ ObCX ; hence τ induces a pretopology τx on CXx . In particular, a structure EX of a right exact category on CX induces a structure EXx of a right exact category on CXx . If (CX , EX ) has enough projectives, then spl (CXx , EXx ) has enough projectives. Finally, if the class EX of split epimorphisms of CX is stable under base change, then the class Espl of split epimorphisms of C has this Xx Xx property.

130 C1.9. Monads on categories with an initial object and monads on corresponding pointed categories.

C1.9.1. Deﬁnition. Fix an object (CX , x) of the category Cat?.A monad on φ (CX , x) as a pair(F, φ), where F = (F, µ) is a monad on CX and F (x) −→ x is an F-module structure on the initial object x. We denote by Mon(CX , x) the category whose objects are monads on (CX , x); mor- g phisms from (F, φ) to (F 0, φ0) are monad morphisms F −→ F 0 such that φ = φ0 ◦ g(x).

C1.9.2. Lemma. Every monad (F, φ) on (CX , x) deﬁnes a monad Fφ = (Fφ, µφ) on the corresponding pointed category CXx . The map (F, φ) 7−→ Fφ extends to an isomorphism between the category Mon(CX , x) of monads on CXx and the category Mon(CXx ) of monads on CXx . Proof is left to the reader.

C1.9.3. A remark on augmented monads. Every augmented monad (F, ) on the category CX (see C1.4.1) deﬁnes a monad (F, (x)) on (CX , x), hence a monad on the associated pointed category CXx . The map (F, ) 7−→ (F, (x)) is functorial; so that we have functors + ∼ Mon (CX ) −→ Mon(CX , x) −→ Mon(CXx ).

On the other hand, it is easy to see that there is a natural isomorphism between + + the category Mon (CX ) of augmented monads on CX and the category Mon (CXx ) of augmented monads on the pointed category CXx .

C2. Diagram chasing.

C2.1. Proposition. Let (CX , EX ) be a right exact category with an initial object x and kernels of morphisms; and let

x  y 0 0 β1 α1 Ker(f 0) −−−→ Ker(f) −−−→ Ker(f 00)    0   00 k y k y y k (1) β α 0 1 1 00 A −−−→ A1 −−−→ A 1  1 0   00 f y f y y f β α 0 2 2 00 x −−−→ A2 −−−→ A2 −−−→ A2 be a commutative diagram whose two lower rows and the right column are ’exact’. Then 0 0 β1 α1 its upper row, Ker(f 0) −−−→ Ker(f) −−−→ Ker(f 00), is ’exact’.

131 Proof. Let x be an initial object of CX ; and let Ker(β2) = x. The diagram (1) gives rise to the comutative diagram

0 0 0 e1 j1 α1 Ker(f 0) −−−→ Ker(fe) −−−→ Ker(f) −−−→ Ker(f 00)     0    00 k y cart ek y cart k y y k e j α 0 1 1 1 00 (1.1) A −−−→ Ker(α1) −−−→ A1 −−−→ A 1   1 0    00 f y fe y f y y f id β α 0 0 2 2 00 A2 −−−→ Ker(α2) = A2 −−−→ A2 −−−→ A2

0 0 0 0 β2 where e1 is a deflation, j1 ◦ e1 = β1, j1 ◦ e1 = β1, and A2 −→ A2 is the kernel of α2. The e0 1 0 claim is that the morphism Ker(f) −−−→ Ker(α1) is a deflation. By 2.3.4.1 (or 2.3.4.3), the upper left square of (1.1) is cartesian, because the left e 0 1 lower horizontal arrow is identical. Since A1 −−−→ Ker(α1) is a deflation, this implies 0 e1 that Ker(f 0) −−−→ Ker(fe) is a deflation.

Since Ker(β2) is trivial, it follows from 2.3.4.3 (applied to the middle section of the diagram (1.1)) that the upper middle square of (1.1) is cartesian.

0 0 j1 α1 Notice that Ker(fe) −−−→ Ker(f) is the kernel of Ker(f) −−−→ Ker(f 00). k00 00 00 In fact, by hypothesis, the kernel of Ker(f ) −−−→ A1 is trivial. Therefore, by 3.3.4.3, the right square of the commutative diagram

0 0 j1 α1 Ker(α0 ) −−−→ Ker(f) −−−→ Ker(f 00)  1   0   00 ek y cart k y y k j α 1 1 00 Ker(α1) −−−→ A1 −−−→ A1

0 is cartesian (whenever Ker(α1) exists). Therefore, by the universality of cartesian squares, ∼ 0 there is a natural isomorphism Ker(fe) −→ Ker(α1).

The following assertion is a non-additive verstion of the ’snake lemma’. Its proof is not reduced to the element-wise diagram chasing, like the argument of the classical ’snake lemma’. Therefore, it requires more elaboration than its abelian prototype.

C2.2. Proposition (’snake lemma’). Let (CX , EX ) be a right exact category with

132 an initial object x; and let

x  y 0 0 β1 α1 Ker(f 0) −−−→ Ker(f) −−−→ Ker(f 00)    0   00 k y k y y k β α 0 1 1 00 A −−−→ A1 −−−→ A (2) 1  1 0   00 f y f y y f β α 0 2 2 00 x −−−→ A −−−→ A2 −−−→ A 2  2 0   00 e y e y y e β α 0 3 3 00 A3 −−−→ A3 −−−→ A3 be a commutative diagram whose vertical columns and middle rows are ’exact’, the arrows k00 0 00 00 00 α1, e , e, e are deﬂations, and the kernel of Ker(f ) −−−→ A1 is trivial. (a) Suppose that each deﬂation of (CX , EX ) is isomorphic to its coimage and the 00 unique arrow x −→ A3 is a monomorphism. Then there exists a natural morphism 00 d 0 Ker(f ) −→ A3 such that the sequence 0 0 β1 α1 Ker(f 0) −−−→ Ker(f) −−−→ Ker(f 00)   y d (3) β α 0 3 3 00 A3 −−−→ A3 −−−→ A3

0 0 β1 α1 is a complex. Moreover, its subsequences Ker(f 0) −−−→ Ker(f) −−−→ Ker(f 00) and d β 00 0 2 Ker(f ) −−−→ A3 −−−→ A3 are ’exact’. (b) Suppose, in addition, that (b1) EX is saturated in the following sense: if λ◦s is a deflation and s is a deflation, then λ is a deflation; (b2) the following condition holds: p (#) If M −→e N is a deflation and M −→ M an idempotent morphism (i.e. p2 = p) which has a kernel and such that the composition e◦p is a trivial morphism, then the composition k(p) of the canonical morphism Ker(p) −−−→ M and M −→e N is a deflation. Then the entire sequence (3) is ’exact’.

Proof. (i) Since α1 is a deﬂation, there exists a cartesian square

α e1 00 Ae1 −−−→ Ker(f )   00  00 ek y cart y k α 1 00 A1 −−−→ A1

133 00 where αe1 is a deﬂation too. It follows from 2.3.4.1 that Ae1 = Ker(α2f) = Ker(f α1). This is seen from the commutative diagram

id α 00 e1 00 Ae1 −−−→ Ker(f α1) −−−→ Ker(f )     00   00 id y ek y cart y k k00 α e 1 00 Ker(α2f) −−−→ A1 −−−→ A   1    00 (4) h y cart f y y f β α 0 2 2 00 A −−−→ A2 −−−→ A 2  2 0   00 e y e y y e β α 0 3 3 00 A3 −−−→ A3 −−−→ A3 with cartesian squares as indicated. (ii) By 2.3.3, we have a commutative diagram

k(α1) α e e1 00 Ker(α1) −−−→ Ae1 −−−→ Ker(f ) e    00  00 oy ek y cart y k k(α1) α 1 00 Ker(α1) −−−→ A1 −−−→ A1 whose (rows are conﬂations and the) left vertical arrow is an isomorphism. Thus, we obtain a commutative diagram

k(α1) α e e1 00 Ker(α1) −−−→ Ae1 −−−→ Ker(f ) x e    00  00 e1 ek y cart y k β α 0 1 1 00 A −−−→ A1 −−−→ A 1  1 0   00 (5) f y f y y f β α 0 2 2 00 A −−−→ A2 −−−→ A 2  2 0   00 e y e y y e β α 0 3 3 00 A3 −−−→ A3 −−−→ A3

Since the second row of the diagram (2) is ’exact’, the morphism e1 is a deﬂation. (iii) Combining the diagram (4) with (the left upper corner of) (5), we obtain a

134 commutative diagram

e k(α1) α 0 1 e 00 e1 00 A −−−→ Ker(α1) −−−→ Ker(f α1) −−−→ Ker(f ) 1  e    00   00 k(αe1)y ek y cart y k k00 α e 1 00 Ker(α2f) −−−→ A1 −−−→ A   1    00 (6) h y cart f y y f β α 0 2 2 00 A −−−→ A2 −−−→ A 2  2 0   00 e y e y y e β α 0 3 3 00 A3 −−−→ A3 −−−→ A3

00 00 0 where ek ◦k(αe1)◦e1 = β1. Therefore, β2 ◦(h◦k(αe1)◦e1) = f ◦(ek ◦k(αe1)◦e1) = f ◦β1 = β2 ◦f . 0 Since the left middle square of (6) is cartesian, this implies that h ◦ k(αe1) ◦ e1 = f . 0 0 0 Therefore, e ◦ h ◦ k(αe1) ◦ e1 = e ◦ f is a trivial morphism. 0 (iv) Notice that, by 2.1.2, the kernel morphism Ker(e ◦ h ◦ k(αe1))−−−→Ker(αe1) is a 0 0 monomorphism, because A3 has a morphism to x, hence x −→ A3 is a (split) monomorphism. Since e1 is a deflation, in particular a strict epimorphism, it follows from 2.3.4.4 that 0 the composition (e ◦h)◦k(αe1) is trivial. By hypothesis, αe1 (being a deflation) is isomorphic 00 to the coimage morphism, i.e. Ker(f ) is naturally isomorphic to Coim(αe1). Therefore, 0 0 the morphism e ◦ h factors through αe1, i.e. e ◦ h = d ◦ αe1. Since αe1 is a deflation, in 00 d 0 particular an epimorphism, the latter equality determines the morphism Ker(f ) −→ A3 uniquely. 0 0 β1 α1 (v) By C2.1, the sequence Ker(f 0) −−−→ Ker(f) −−−→ Ker(f 00) is ’exact’. α0 d 1 00 00 0 (vi) The composition of Ker(f) −−−→ Ker(f ) and Ker(f ) −−−→ A3 is trivial. In fact, the diagram (6) induces a commutative diagram

0 ∼ α1 A0 Ker(h) −−−→ Ker(f) −−−→ Ker(f 00) 1        00 e1y k(h)y k y y k 00 k(α1) k α e e 1 00 Ker(α1) −−−→ Ker(α2f) −−−→ A1 −−−→ A e   1    00 (7) h y cart f y y f β α 0 2 2 00 A −−−→ A2 −−−→ A 2  2 0   00 e y e y y e β α 0 3 3 00 A3 −−−→ A3 −−−→ A3 where the isomorphism Ker(h) −→∼ Ker(f) is due the fact that the left middle square of the diagram (7) is cartesian. We can and will assume that this isomorphism is identical. α0 k(h) 1 00 The morphism Ker(f) −−−→ Ker(f ) is the composition of Ker(f) −−−→ Ker(α2f) and

135 α e1 00 0 0 Ker(α2f) −−−→ Ker(f ). Therefore, d ◦ α1 = d ◦ αe1 ◦ k(h) = e ◦ h ◦ k(h), which shows 0 that the composition d ◦ α1 is trivial, because already the composition h ◦ k(h) is trivial. (vii) The argument above can be summarized in the commutative diagram

γ A0 Ker(f) −−−→ Ker(d) 1      e1y y k(h) y k(d)

k(α1) α e e1 00 (8) Ker(α1) −−−→ Ker(α2f) −−−→ K(f )  e      e2y h y y d k(e0) e0 0 0 0 Ker(e ) −−−→ A2 −−−→ A3

e 2 0 where Ker(αe1) −−−→ Ker(e ) is a deﬂation. Taking into consideration the cartesian square

γ M −−−→e Ker(d)     µ y y k(d) (9) α e1 00 Ker(α2f) −−−→ K(f ) we extend (8) to the commutative diagram

id γ Ker(f) −−−→ Ker(h) −−−→ Ker(d)       id y k(eh) y y id k(eh) eγ A0 Ker(eh) −−−→ M −−−→ Ker(d) 1        e1y k(eh)y µ y cart y k(d) (10) k µ α 1 e1 00 Ker(α1) −−−→ M −−−→ Ker(α2f) −−−→ K(f )  e        e2y eh y cart h y y d id k(e0) e0 0 0 0 0 Ker(e ) −−−→ Ker(e ) −−−→ A2 −−−→ A3 where µ ◦ k(eh) = k(h), and µ ◦ k1 = k(αe1). Since the square (9) is cartesian and αe1 is a deﬂation, its pull-back, γe, is a deﬂation too. Notice that the commutativity of the left lower square and the fact that e2 is a strict epimorphism imply that eh is a strict epimorphism. Consider the cartesian square

0 e3 M −−−→ M f    p y y eh (11) e 0 3 0 A1 −−−→ Ker(e )

136 0 e3 where e3 = e2 ◦ e1. Since e3 is a deflation, the arrow Mf −−−→ M is a deflation. Since 0 s eh ◦ k1 ◦ e1 = e3, the projection p has a splitting, A1 −→ Mf; i.e. p ◦ s = id. Set p = s ◦ p and 0 p ϕ = γe ◦ e3. It follows that Mf −→ Mf is an idempotent (– a projector), ϕ is a deflation, the 0 composition ϕ◦p = γe◦(e3 ◦s)◦p = γe◦(k1 ◦e1)◦p is trivial, because k(d)◦γe◦k1 = αe1 ◦k(αe1) k(d) is trivial and Ker(d) −−−→ K(f 00) is a monomorphism. The latter follows from the fact 0 0 that A3 has a morphism to x, hence the unique arrow x −→ A3 is a (split) monomorphism. Since the square (11) is cartesian, it follows from 2.3.4.1 that Ker(p) is naturally isomorphic to Ker(eh) = Ker(f). And, by 2.3.4.3, Ker(p) is naturally isomorphic to Ker(p), because p = s ◦ p and s is a monomorphism. Thus, Ker(p) is naturally isomorphic to Ker(f). (viii) Suppose that the condition (#) of the proposition holds. Then the composition of ϕ Ker(p) −→ Mf and Mf −→ Ker(d) is a deflation; hence the composition of the morphisms k(eh) γ γ Ker(f) −−−→ M and M −−−→e Ker(d) (i.e. the morphism Ker(f) −−−→ Ker(d) in the diagram (10)) is a deflation. k(β ) 3 0 (ix) The claim is that d is the composition of the morphism Ker(β3) −−−→ A3 and d0 00 a deflation Ker(f ) −−−→ Ker(β3). Since αe1 is a deflation, it suffices to prove a similar 0 assertion for d ◦ αe1 = e ◦ h. We have a commutative diagram

k00 α e 1 00 B −−−→ A1 −−−→ A   1    00 th y cart y tf y t

β3 λ B −−−→e Ker(e) −−−→ Ker(e00)       00 (12) ψ y cart y k(e) y k(e ) ψ β α 0 2 2 00 B −−−→ A −−−→ A2 −−−→ A  2  2 0 0   00 e y cart e y y e y e k(β3) β α 0 3 3 00 Ker(β3) −−−→ A3 −−−→ A3 −−−→ A3

00 00 00 where f = k(e) ◦ tf , f = k(e ) ◦ t and the remaining new arrows are determined by the commutativity of the diagram (12) and by being a part of a cartesian square. By hypothesis, the columns of the diagram (2) are ’exact’; in particular, the morphism tf is a deﬂation. Therefore, the morphism B −→th B is a deﬂation. Being the composition of two cartesian diagrams, the diagram

k00 B −−−→e A  1   ψ ◦ thy y k(e) ◦ tf = f β 0 2 A2 −−−→ A2

137 is cartesian, as well as the diagram

00 ek Ker(α2f) −−−→ A1     h y y f β 0 2 A2 −−−→ A2 Therefore, they are isomorphic to each other. So, we can and will assume that B = Ker(α2f) and h = ψ ◦ th. It follows from (the left part of) the diagram (12) that

0 0 0 e ◦ h = e ◦ ψ ◦ th = k(β3) ◦ (e ◦ th),

k(β ) 3 0 0 that is e ◦ h is the composition of Ker(β3) −−−→ A3 and the deflation e ◦ th. 0 (x) The composition α3 ◦β3 is trivial by 2.3.4.4, because the composition α3 ◦β3 ◦e = 00 00 0 e ◦ α2 ◦ β2 is trivial, x −→ A3 is a monomorphism (by hypothesis), and e is a deflation, hence a strict epimorphism. The claim is that, if (CX , EX ) has the propery (#), then the β k 0 3 3 00 morphism A3 −−−→ A3 is the composition of the kernel morphism Ker(α3) −−−→ A3 and t 0 3 a deflation A3 −−−→ Ker(α3). 00 Since in the upper right square of the diagram (12), the arrows α1, t , and tf are defla- λ tions, the forth arrow, Ker(e) −−−→ Ker(e00), is a deflation too (due to the saturatedness condition (b1)). Consider the commutative diagram

λ Ker(e) −−−→ Ker(e00)     v y y id 0 t3 p2 A0 −−−→ D −−−→ Ker(e00) 2   0  0   00 t3 y β2 y cart y k(e ) (13) t0 β0 α 0 3 2 2 00 A −−−→ D −−−→ A2 −−−→ A 2   2 0    00 e y u y cart e y y e t k(α3) α 0 3 3 00 A3 −−−→ Ker(α3) −−−→ A3 −−−→ A3 0 0 0 where β2 ◦ v = k(e), β2 ◦ t2 = β2. The upper left corner of the commutative diagram (13) gives rise to the commutative diagram t0 p 0 e3 e2 Ae −−−→ De −−−→ Ker(e) 2   0    λ y cart λe y cart y λ (14) t0 p 0 3 2 00 A2 −−−→ D −−−→ Ker(e ) whose both squares are cartesian. Since λ is a deﬂation, all vertical arrows of (14) are v deﬂations, as well as the arrows p2 and ep2. The morphism Ker(e) −−−→ D determines a

138 s2 splitting Ker(e) −−−→ De of the projection ep2. Let p2 denote the composition s2 ◦ p2. It follows that p2 is an idempotent De −→ De and the composition

0 0 k(α3) ◦ (u ◦ λe) ◦ p2 = e ◦ β2 ◦ (λe ◦ s2) ◦ ep2 = e ◦ β2 ◦ v ◦ ep2 = (e ◦ k(e)) ◦ ep2 is trivial. Therefore, (u ◦ λe) ◦ p2 is trivial. The kernel of the idempotent p2 is isomorphic to the kernel of ep2. Since the right square of (14) is cartesian, there is a natural isomorphism Ker(p2) ' Ker(ep2). It follows from the right cartesian square of (13) that there is a 0 natural isomorphism Ker(p2) ' Ker(α2) = A2. If the right exact category (CX , EX ) has the property (#), then the above implies that u◦t0 0 3 0 0 0 the morphism A2 −−−→ Ker(α3) is a deflation. Since u ◦ t3 = t3 ◦ e and e is a deflation, t 0 3 the morphism A3 −−−→ Ker(α3) is a deflation. C2.3. Remarks about conditions of the ’snake lemma’. Fix a right exact category (CX , EX ). The main condition of the ’snake lemma’ C2.2, the one which garantees the existence of the connecting morphism d, is that each deflation M −→e N is isomorphic c(e) to its coimage morphism M −−−→ Coim(e) = M/Ker(e). If the category CX is additive, then every strict epimorphism which has a kernel, in particular, every deflation, is isomorphic to its coimage morphism. The latter property holds in many non-additive categories, for instance in the category Algk of unital associative k-algebras (see 2.3.5.3). Similarly, the property p (#) If M −→e N is a deflation and M −→ M an idempotent morphism (i.e. p2 = p) which has a kernel and such that the composition e ◦ p is a trivial morphism, then the k(p) composition of the canonical morphism Ker(p) −−−→ M and the deflation M −→e N is a deflation. which ensures ’exactness’ of the ’snake’ sequence (3) holds in any additive category. In fact, if the category CX is additive, then the existence of the kernel of p means q precisely that the idempotent q = idM − p is splittable; i.e. M −→ M is the composition k(p) of Ker(p) −−−→ M and a (strict) epimorphism M −→t Ker(p) such that t ◦ k(p) = id. The condition e◦p is trivial (that is e◦p = 0) is equivalent to the equalities e = e◦q = (e◦k(p))◦t which imply (under saturatedness condition, cf. C2.2(b1)) that e ◦ k(p) is a deflation.

C2.3.1. Example. The property (#) holds in the category Algk. In fact, let ϕ p A −→ B be a strict algebra epimorphism, and A −→ A an idempotent endomorphism such that the composition ϕ ◦ p is a trivial morphism; that it equals to the composition of an augmentation morphism A −→π k and the k-algebra structure k −→iB B. In particular, A = k ⊕ A+, where A+ = K(π) is the kernel of the augmentation π in the usual sense. On the other hand, Ker(p) = k ⊕ K(p), and, since p ◦ p = p and the ideal K(p) def= {y ∈ A | p(y) = 0} coincides {x − p(x) | x ∈ A}. Similarly, Ker(ϕ ◦ p) = k ⊕ K(ϕ ◦ p), and it follows that K(ϕ ◦ p) = A+.

139 Every element x of A is uniquely represented as λ · 1A + x+, where 1A is the unit element of the algebra A and x+ ∈ A+. Therefore, x − p(x) = x+ − p(x+) and

ϕ(µ · 1A + (x − p(x))) = µ · 1B + ϕ(x+ − p(x+)) = µ · 1B + ϕ(x+) = ϕ(µ · 1A + x+).

Since µ ∈ k and x+ ∈ A+ are arbitrary and ϕ is a strict epimorphism (that is a surjective map), this shows that ϕ ◦ k(p) is a strict epimorphism.

C3. Localizations of exact categories and (co)quasi-suspended categories. t-Structures.

u∗ C3.1. Remarks on localizations. Let CX −→ CZ be a functor. Suppose that the category CZ is cocomplete, i.e. it has colimits of arbitrary small diagrams (equivalently, it has inﬁnite coproducts and cokernels of pairs of arrows). By [GZ, II.1], the functor u∗ hX equals to the composition of the Yoneda embedding CX −→ CbX of the category CX into the category CbX of presheaves of sets on CX and a continuous (that is having a right u∗ adjoint) functor CbX −→e CZ . Since every presheaf of sets on a category is a colimit of a ∗ canonical diagram of representable presheaves and the functor ue preserves colimits, it is determined uniquely up to isomorphism. q∗ In particular, every functor CX −→ CY gives rise to a commutative diagram

q∗ C −−−→ C X Y   hX y y hY (1) ∗ bq CbX −−−→ CbY

∗ with a continuous functor qb determined by the commutativity of (1) uniquely up to isomorphism. C3.1.1. Lemma. (a) The functor

q b∗ ∗ CbY −−−→ CbX ,F 7−→ F ◦ q , (2)

∗ is a canonical right adjoint to qb . ∗ (b) If the functor q has a right adjoint q∗, then the diagram

q∗ C ←−−− C X Y h   h X y y Y (1∗) bq∗ CbX ←−−− CbY quasi-commutes.

140 ∗ Proof. (a) Recall that the functor qb is determined uniquely up to isomorphism by the ∗ ∗ equality qb (hY (L)) = hY (q (L)) for all L ∈ ObCX . For every L ∈ ObCX and F ∈ ObCbY , we have

∗ ∗ ∗ ∗ CbX (hX (L),F ◦ q ) ' F (q (L)) ' CbY (hY (q (L)),F ) ' CbY (qb (hY (L)),F ). Since all isomorphisms here are functorial, it follows that the functor (2) is a right ∗ adjoint to qb . (b) For any L ∈ ObCY ,

∗ ∗ qb∗(hY (L)) = hY (L) ◦ q = CY (q (−),L) ' CX (−, q∗(L)) = hX (q∗(L)), hence the assertion. q∗ C3.1.1.1. Corollary. For every functor CX −→ CY , the functor qb∗ has a right ! adjoint, qb . In particular, qb∗ is exact. Proof. The fact follows from C3.1.1(a).

q∗ C3.1.2. Proposition. If CX −→ CY is a localization, then the continuous functor ∗ qb in (1) is a localization too. ∗ q∗ q bf Proof. The functor CbX −→b CbY is decomposed into a localization CbX −→ CZ at q∗ Σ = {s ∈ HomC | q∗(s) is invertible} and a conservative functor C −→bc C . Since q∗ bX b Z bY ∗b ∗ q is a localization and the composition qbf ◦ hX makes invertible all arrows of Σq∗ = {s ∈ ∗ Ψ HomCbX | q (s) is invertible}, there exists a unique functor CY −→ CZ such that the diagram q∗ C −−−→ C X Y   hX y y Ψ (2) q∗ bf CbX −−−→ CZ ∗ commutes. The localization qbf is continuous, i.e. it has a right adjoint which is, forcibly, a fully faithful functor. Therefore, by [GZ, I.1.4], the category CZ has limits and colimits Ψ of arbitrary (small) diagrams. Therefore, the functor CY −→ CZ is the composition of the 0 hY Ψ Yoneda imbedding CY −→ CbY and a continuous functor CbY −→ CZ ; the latter is deﬁned uniquely up to isomorphism. Thus, we have the equalities

∗ ∗ 0 ∗ 0 ∗ 0 ∗ ∗ qbf ◦ hX = Ψ ◦ q = Ψ ◦ hY ◦ q = Ψ ◦ qb ◦ hX = (Ψ ◦ qbc ) ◦ qbf ◦ hX ∗ 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ (3) (qbc ◦ Ψ ) ◦ hY ◦ q = qbc ◦ Ψ ◦ q = qbc ◦ qbf ◦ hX = qb ◦ hX ' hY ◦ q

∗ 0 ∗ ∗ The equality qbf ◦ hX = (Ψ ◦ qbc ) ◦ qbf ◦ hX implies, thanks to the continuity of the functors 0 ∗ ∗ ∗ Ψ ◦ qbc and qbf and the universal properties of the localization qbf , that the composition 0 ∗ Ψ ◦ qbc is isomorphic to the identity functor. 141 Similarly, thanks to the universal properties of the localization q∗, the isomorphism ∗ 0 ∗ ∗ ∗ 0 ∗ 0 (qbc ◦ Ψ ) ◦ hY ◦ q ' hY ◦ q implies that (qbc ◦ Ψ ) ◦ hY ' hY . Since the functor qbc ◦ Ψ is continuous and every presheaf of sets on CY is a colimit of a (canonical) diagram of ∗ 0 representable presheaves, it follows from the latter isomorphism that the composition qbc ◦Ψ ∗ 0 is isomorphic to the identical functor. All together shows that qbc and Ψ are mutually quasi-inverse category equivalences.

q∗ C3.1.3. Note. Suppose that CX and CY are k-linear categories and CX −→ CY a k-linear functor. If the category CY is cocomplete, then it follows from the assertion [GZ, II.1] mentioned above that there exists a unique up to isomorphism continuous func- ∗ bq ∗ ∗ tor Mk(X) −→ CY such that q = qb ◦ hX . Here, as above, Mk(X) is the category of k-presheaves on the category CX . This establishes an equivalence between the cate- c gory Homk(CX ,CY ) of k-linear functors CX −→ CY and the category Homk(CX ,CY ) of continuous k-linear functors Mk(X) −→ CY . q∗ If a k-linear functor CX −→ CY is equivalent to a localization functor (i.e. it is the def ∗ composition of the localization functor at Σq∗ = {s ∈ HomCX | q (s) is invertible} and −1 a category equivalence Σq∗ CX −→ CY ), then the argument of C3.1.1 with the categories of presheaves of sets replaced by the categories of k-presheaves shows that the natural q∗ extension Mk(X) −→b Mk(Y ) is equivalent to a continuous localization.

C3.2. Right weakly ’exact’ functors and ’exact’ localizations. Let (CX , EX ) and (CY , EY ) be exact categories. A right weakly ’exact’ functor (CX , EX ) −→ (CY , EY ) is a ϕ∗ j e functor CX −→ CY such that for every conﬂation L −→ M −→ N, there is a commutative diagram ϕ∗(j) ϕ∗(e) ϕ∗(L) −−−→ ϕ∗(M) −−−→ ϕ∗(N) e0 &% j0 L1

j0 ϕ∗(e) 0 ∗ ∗ in which e is a deflation and L1 −−−→ ϕ (M) −−−→ ϕ (N) is a conflation. ∗ jX Recall that the Gabriel-Quillen embedding CX −→ CXE is the composition of the ∗ hX qX Yoneda embedding CX −→ Mk(X) and the sheafification functor Mk(X) −→ CXE .

C3.2.1. Proposition. Let (CX , EX ) and (CY , EY ) be exact k-linear categories and ϕ∗ (CX , EX ) −−−→ (CY , EY ) a right ’exact’ k-linear functor. ∗ ϕe (a) There is a unique up to isomorphism continuous k-linear functor CXE −−−→ CYE such that the diagram ϕ∗ C −−−→ C X Y ∗   ∗ jX y y jY ∗ ϕe CXE −−−→ CYE commutes. Here the vertical arrows are the Gabriel-Quillen embeddings.

142 ∗ ∗ ϕ ϕe (b) If the functor CX −→ CY is a localization, then the functor CXE −→ CYE is a localization. (c) Suppose that the following condition holds: for every L ∈ ObCX and every deﬂation N −→e ϕ∗(L), there exist a deﬂation M −→t L and a commutative diagram

g ϕ∗(M) −−−→ N ϕ∗(t) &. e ϕ∗(L)

ϕ Then a right adjoint C −→e∗ C to the functor ϕ∗ has a right adjoint, ϕ!. In YE XE e e particular, the functor ϕe∗ is exact.

Proof. (a) Objects of the category CXE – k-sheaves on the pretopology (CX , EX ), are naturally identiﬁed with right ’exact’ k-linear functors from CX to the abelian category ∗ op ϕ Mk(X) . Therefore, since the functor CX −→ CY is right ’exact’, the composition with ϕb∗ it maps CYE to CXE . By C3.1.1, we can (and will) assume that the functor Mk(Y ) −→ ∗ Mk(X) is given by F 7−→ F ◦ ϕ . Thus, we have a commutative diagram

ϕb∗ Mk(X) ←−−− Mk(Y ) x x   qX∗   qY ∗ ϕe∗ CXE ←−−− CYE whose vertical arrows are inclusion functors. This diagram yields, by adjunction, a quasi- commutative diagram ∗ ϕb Mk(X) −−−→ Mk(Y )   ∗   ∗ qX y y qY (1) ∗ ϕe CXE −−−→ CYE where the vertical arrows are sheafification functors. The sheafification functors are exact ∗ ∗ ∗ ∗ ∗ ∗ ∗ localizations. An isomorphism qY ϕb ' ϕe qX implies that ϕe ' qY ϕb qX∗, because the adjunction arrow Id −→ q∗ q is an isomorphism. Together with the isomorphism CXE X X∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ qY ϕb ' ϕe qX , this implies that the canonical morphism qY ϕb −→ qY ϕb qX∗qX is an def ∗ ∗ isomorphism. The claim is that the functor ϕe∗ = qX ϕb∗qY ∗ is a right adjoint to ϕe . In fact, the composition of morphisms

∼ ∗ ∗ ∗ ∗ ∗ ∗ ∼ ∗ IdC −→ q qY ∗ −→ q ϕ∗ϕ qY ∗ −→ q ϕ∗qX∗q ϕ qY ∗ −→ ϕ∗ϕ YE Y Y b b Y b X b e e and ∗ ∼ ∗ ∗ ∗ ∼ ∗ ∗ ∗ ∼ ϕ ϕ∗ −→ q ϕ qY ∗q ϕ∗qX∗ −→ q ϕ ϕ∗qX∗ −→ q qX∗ −→ IdC e e X b Y b X b b X XE are adjunction arrows.

143 ϕ∗ (b) By C3.1.1 (and C3.1.2), the continuous functor Mk(X) −→b Mk(Y ) is a localization. Thus, the three arrows of the quasi-commutaive diagram (1) are localizations, hence ∗ the forth one, ϕe , is a localization. (c) The condition (c) means that for every L ∈ ObCY and every presheaf F of k- ∗ modules on CX , the value of the associated sheaf qX (F ) at ϕ (L) can be computed using only deﬂations (– covers) of the form ϕ∗(M −→t L), where M −→t L is a deﬂation. This implies that the diagram ϕb∗ Mk(X) ←−−− Mk(Y )   ∗   ∗ qX y y qY (1∗) ϕe∗ CXE ←−−− CYE ∗ ! quasi-commutes. Therefore, by the argument similar to (a) above, the functor qY ϕb qX∗ is a right adjoint to ϕe∗.

C3.3. Example. Suppose that CX is a k-linear category with the smallest exact structure (given by split conﬂations). Then any k-linear functor (in particular, any right ϕ∗ or left ’exact’ k-linear functor) CX −→ CY is ’exact’. The category CXE coincides with the category Mk(X) of k-presheaves on CX , and the functor

∗ ϕe CXE = Mk(X) −−−→ CYE

∗ ϕb is isomorphic to the composition of the functor Mk(X) −−−→ Mk(Y ) and the sheaﬁﬁca- ∗ qY tion functor M (Y ) −−−→ C . Therefore, a right adjoint ϕ to ϕ∗ is isomorphic to the k YE e∗ e composition ϕb∗qY ∗, which is not, usually, an exact functor.

C3.3.1. Example. Let (CX , EX ) be an exact k-linear category. Suppose that CY spl is an additive k-linear category endowed with the smallest exact structure, EY . Then a ∗ ϕ spl functor CX −→ CY is right ’exact’ functor from (CX , EX ) to (CX , EY ) iff it maps every deflation of the exact category (CX , EX ) to a split epimorphism (i.e. a coretraction). Notice that the condition (c) of C3.2.1 holds because every deflation in CY splits. Therefore, by ! C3.2.1(c), the functor ϕe∗ has a right adjoint, ϕe . spl If the exact structure on (CX , EX ) is also the smallest one (i.e. EX = EX ), then C = M (X) and C = M (Y ); i.e. in this case ϕ∗ = ϕ∗ and, therefore, a right XE k YE k e b ! adjoint to the functor ϕe∗ coincides with ϕb . ϕ∗ C3.4. Remark. Let (CX , EX ) and (CY , EY ) be exact categories. If CY −→ CX is an arbitrary functor, one can still define functors

∗ ! ϕe ϕe∗ ϕe CYE −−−→ CXE −−−→ CYE −−−→ CXE by the formulas ∗ ∗ ∗ ∗ ! ∗ ! ϕe = qX ϕb qY ∗, ϕe∗ = qY ϕb∗qX∗, ϕe = qX ϕb qY ∗. (2) 144 ∗ ! The functors ϕe , ϕe∗, and ϕe might be regarded as derived functors of respectively ∗ ! ϕb , ϕb∗, and ϕb (this viewpoint is discussed in Section D). C3.5. Proposition. Let (CX , EX ) and (CY , EY ) be exact k-linear categories and ∗ ϕ ∗ (CX , EX ) −→ (CY , EY ) a right ’exact’ k-linear functor. Suppose that ϕ is a localization ∗ ∗ functor. Then ϕ is ’exact’ iff the class of arrows Σϕ∗ = {s ∈ HomCX | ϕ (s) is an isomorphism} satisfies the following condition: (#) If the rows of a commutative diagram L −−−→ M −−−→ N       y y y (2) L0 −−−→ M 0 −−−→ N 0 are conflations and any two of its vertical arrows belong to Σϕ∗ , then the remaining arrow belongs to Σϕ∗ . Proof. (i) Consider first the case when ϕ∗ is the identical functor. Let 0 −−−→ L −−−→ M −−−→ N −−−→ 0       y y y (3) 0 −−−→ L0 −−−→ M 0 −−−→ N 0 −−−→ 0 0 0 0 be a commutative diagram in CY such that L −→ M −→ N and L −→ M −→ N are conflations. If two of the three vertical arrows are isomorphisms, then the third arrow is an isomorphism as well. In fact, the Gabriel-Quillen embedding transforms the diagram (2) into a commutative diagram with exact rows. If two of the vertical arrows of such diagram are isomorphisms, then the third one is an isomorphism. The Gabriel-Quillen embedding is a fully faithful functor, in particular, it is conservative. Therefore, all vertical arrows in the original diagram are isomorphisms. ϕ∗ (ii) Suppose that the functor CX −→ CY is ’exact’; i.e. it maps conflations to confla- ∗ tions. In particular, ϕ maps a diagram (2) with two arrows from Σϕ∗ to a diagram whose rows are conflations and two vertical arrows are isomorphisms. By (i) above, the third arrow is an isomorphism too; i.e. all vertical arrows of the diagram (2) belong to Σϕ∗ . ϕ∗ (iii) Suppose now that CX −→ CY is a localization functor which is right ’exact’ and satisfies the condition (#). The claim is that the functor ϕ∗ is ’exact’. j e ∗ Let L −→ M −→ N be a conflation in CX . The functor ϕ being right ’exact’ means that there is a commutative diagram ϕ∗(j) ϕ∗(e) ϕ∗(L) −−−→ ϕ∗(M) −−−→ ϕ∗(N) e0 &% j0 (4) Le j0 ϕ∗(e) ∗ ∗ 0 ∗ such that Le −−−→ ϕ (M) −−−→ ϕ (N) is a conflation in CY and e ∈ EY . Since ϕ is a ∗ 0 0 0 localization, we can and will assume that Le = ϕ (L ) for some L ∈ ObCX . Let j be the −1 ϕ∗(j00) ϕ∗ (s) ∗ 0 ∗ ∗ ∗ composition of arrows ϕ (L ) −−−→ ϕ (M1) and ϕ (M1) −−−→ ϕ (M) for some s ∈ Σϕ∗ .

145 Consider the cocartesian square

e M −−−→ N     0 s y y s (4) e1 M1 −−−→ N1

∗ By hypothesis, e1 is a deﬂation and ϕ maps (4) to a cocartesian square. The square (4) is embedded into a commutative diagram

j e L −−−→ M −−−→ N    00   0 s y s y y s (5) j1 e1 L1 −−−→ M1 −−−→ N1

0 whose rows are deﬂations. Since the vertical arrows s, s in (5) belong to Σϕ∗ , the remain- 00 ing vertical arrow, s , belongs to Σϕ∗ . ∗ 00 00 The equality ϕ (e1 ◦ j ) = 0 means that e1 ◦ j ◦ t = 0 for some t ∈ Σϕ∗ . Therefore we have a commutative diagram

j e L −−−→ M −−−→ N    00   0 s y y s y s j1 e1 L −−−→ M −−−→ N (6) x1 x 1 1   00 g   j t L00 −−−→ L0

00 g with a uniquely deﬁned L −→ L1. Thus, we have a commutative diagram

ϕ∗(s00) ∗ ∗ ϕ (L) −−−→ ϕ (L1) ∗ (7) e2 &% ϕ (g) ϕ∗(L00)

e0 ∗ ∗ where the arrow e2 is the composition of the deflation ϕ (L) −−−→ ϕ (L) and the isomor- ϕ∗(s00)−1 ϕ∗(s00) ∗ 0 ∗ 00 ∗ ∗ phism ϕ (L ) −−−→ ϕ (L ). Since ϕ (L) −−−→ ϕ (L1) is an isomorphism, it follows from the commutativity of (7) that the arrow e2 is a retraction; in particular it is a strict monomorphism. On the other hand, e2 is a deflation, hence an epimorphism. Therefore, e2 is an isomorphism, which implies that the deflation e0 in the diagram (4) is an isomorphism. Therefore, ϕ∗ maps the deflation L −→j M −→e N to a deflation.

ϕ∗ C3.5.1. Corollary. Let (CX, EX ) and (CY, EY ) be exact categories and CX −→ CY a left ’exact’ functor. Suppose that ϕ∗ is a localization functor. Then the functor ϕ∗ is

146 ∗ ’exact’ iﬀ the class of arrows Σϕ∗ = {s ∈ HomCX | ϕ (s) is an isomorphism} satisﬁes the condition (#) of C3.5. Proof. The assertion is dual to that of C3.5.

φ∗ C3.6. Proposition. Let (CX, EX ) and (CY, EY ) be exact categories, CX −→ CY an ’exact’ functor, and φ∗ CX −−−→ CY ∗ ∗ φs &% φc −1 Σφ∗ CX its canonical decomposition into a localization and a conservative functor. The functors ∗ ∗ φs and φc are ’exact’. −1 Proof. We call a pair of arrows L −→ M −→ N in Σφ∗ CX a conflation if it is isomorphic to the image of a conflation of CX. We leave to the reader verifying that this −1 defines a structure of an exact category on the quotient category Σφ∗ CX. It follows that ∗ ∗ the functors φs and φc are ’exact’.

C3.7. Proposition. Let (CX, EX) be an exact svelte category, S a family of arrows of CX; and let ExS((CX, EX), −)) be the pseudo-functor which assigns to every exact category (CY, EY) the category of ’exact’ functors from (CX, EX) to (CY, EY) mapping every arrow of S to an isomorphism. The pseudo-functor ExS((CX, EX), −) is representable.

Proof. Let FS be the family of all ’exact’ functors which map S to isomorphisms, and let S¯ denote the family of all arrows which are transformed into isomorphisms by all functors from FS. Since the category CX is svelte, there exists a subset Ω of FS such that the family of all arrows of CX made invertible by functors of Ω coincides with S¯. The product of any set of exact categories is an exact category. In particular, the product CXΩ of targets of functors of Ω is an exact category and the canonical functor FΩ CX −→ CXΩ is an ’exact’ functor. By C3.6, the functor FΩ factors through an ’exact’ FS localization CX −−−→ CS¯−1X. The ’exact’ functor FS is the universal arrow representing the pseudo-functor ExS((CX, EX), −). By the reason of C4.3.4, we need versions of the facts above for exact categories with actions.

C3.7.1. Proposition. Let (CX, EX) be an exact Z+-category, S a family of arrows of CX; and let ExS((CX, EX), −)) be the pseudo-functor which assigns to every exact Z+-category (CY, EY) the category of ’exact’ Z+-functors from (CX, EX) to (CY, EY) mapping every arrow of S to an isomorphism. The pseudo-functor ExS((CX, EX), −) is representable. C3.8. Multiplicative systems in quasi-(co)suspended categories. Fix a quasi- cosuspended category T−CX = (CX, θX, T rX). We call a class Σ of arrows of CX a multiplicative system of the quasi-cosuspended category T−CX if it is θX-invariant, closed under composition, contains all isomorphisms, and satisﬁes the following condition:

147 (L1) for every pair of triangles

0 0 0 d g f 0 d 0 g 0 f 0 θX (L) −→ N −→ M −→ L and θX (L ) −→ N −→ M −→ L and a commutative diagram f M −−−→ L     t y y s f 0 M 0 −−−→ L0 where s and t are elements of Σ, there exists a morphism N −→u N 0 in Σ such that (u, t, s) is a morphism of triangles, i.e. the diagram

d g f θX (L) −−−→ N −−−→ M −−−→ L         θX(s) y u y t y y s d0 g0 f 0 0 0 0 0 θX (L ) −−−→ N −−−→ M −−−→ L commutes. We denote by SM−(X) the preorder (with respect to the inclusion) of all multiplicative s systems and by SM−(X) the preorder of saturated multiplicative systems of the quasi- cosuspended category T−CX. Recall that a multiplicative system Σ in CX is saturated iﬀ the following condition holds: if α, β, γ are arrows of CX such that the compositions αβ and βγ belong to Σ, then β ∈ Σ (equivalently, all three arrows belong to Σ).

C3.8.1. Proposition. (a) Let T−CX = (CX, θX, T rX) and T−CY = (CY, θY, T rY) F be quasi-cosuspended categories and T−CX −→ T−CY a triangle functor. The family of arrows ΣF = {s ∈ HomCX | F (s) is invertible} is a saturated multiplicative system in T−CX. (b) Let T−CX = (CX, θX, T rX) be a quasi-cosuspended category, (CZ , EZ ) an exact H category and T−CX −→ (CZ , EZ ) a homological functor. Then

n ΣH,θX = {s ∈ HomCX | HθX(s) is invertible for all n ≥ 0} is a saturated multiplicative system in T−CX.

Proof. (a) For any functor F , the family ΣF is closed under composition and contains all isomorphisms. The θX-invariance of ΣF and the property (L1) follow from the axioms of quasi-cosuspended categories.

(b) The system ΣH,θX is closed under composition, contains all isomorphisms, and is θX-invariant by construction. It remains to verify the property (L1). Let

0 0 0 d g f 0 d 0 g 0 f 0 θX (L) −→ N −→ M −→ L and θX (L ) −→ N −→ M −→ L

148 be a pair of triangles and f M −−−→ L     t y y s f 0 M 0 −−−→ L0 a commutative diagram with s and t elements of ΣH,θX . By the property (S3) of quasi- cosuspended categories, there exists a morphism N −→u N 0 in Σ such that (u, t, s) is a morphism of triangles, i.e. the diagram

d g f θX (L) −−−→ N −−−→ M −−−→ L         θX(s) y u y t y y s (1) d0 g0 f 0 0 0 0 0 θX (L ) −−−→ N −−−→ M −−−→ L

H commutes. Let H denote the composition of the homological functor CX −→ CZ with the Gabriel-Quillen embedding CZ −→ CZE , we obtain for every nonnegative integer n a commutative diagram

n n n HθX (d) HθX (g) HθX (f) Hθn+1(L) −−−→ Hθn (N) −−−→ Hθn (M) −−−→ Hθn (L) X  X  X  X n+1  n  n   n HθX (s) yo HθX (u) y HθX (t) yo oy HθX (s) Hθn (d0) Hθn (g0) Hθn (f 0) n+1 0 X n 0 X n 0 X n 0 HθX (L ) −−−→ HθX (N ) −−−→ HθX (M ) −−−→ HθX (L ) (2) in the abelian category CZE whose rows are exact sequences and three of the for vertical n arrows are isomorphisms. Therefore the fourth vertical arrow, HθX (u) is an isomorphism for all n ≥ 0; i.e. u belongs to ΣH,θX .

C3.8.2. Proposition. (a) Let T−CX = (CX, θX, T rX) and T−CY = (CY, θY, T rY) F be quasi-cosuspended categories. Every triangle functor T−CX −−−→ T−CY is uniquely Fs represented as the composition of a triangle localization T−CX −−−→ T−CXs and a con- Fc servative triangle functor T−CXs −−−→ T−CY. (b) Let T−CX = (CX, θX, T rX) be a quasi-cosuspended category and (CZ , EZ ) an exact H category. Every homological functor T−CX −−−→ (CZ , EZ ) is uniquely represented as the Hs composition of a triangle localization T−CX −−−→ T−CXs and a conservative homological Hc functor T−CXs −−−→ (CZ , EZ ).

Proof. Let Σ denote the multiplicative system ΣF of C3.8.1(a), or ΣH,θX of C3.8.1(b). −1 Then the quotient category Σ CX is an additive k-linear category having a unique structure (θ,e T rΣ−1X) of a quasi-cosuspended category such that the localization functor

q∗ Σ −1 CX −−−→ Σ CX = CΣ−1X

149 is a strict triangle functor. Here strict means that the quasi-cosuspension functor θe = θΣ−1X ∗ ∗ − is uniquely determined by the equality θe ◦ qΣ = qΣ ◦ θX, and T rΣ−1X is the class of all sequences θe(L) −→ N −→ M −→ L in CΣ−1X which are isomorphic to the images of ∗ triangles of T rX by the localization functor qΣ. Details are left to the reader. − C3.8.3. Proposition. Let T−CX = (CX, θX, TrX) be a svelte quasi-cosuspended category, S a family of arrows of the category CX, and T rS(T−CX, −) the pseudo-functor which assigns to every quasi-cosuspended category T−CY the category of all triangular functors F from T−CX to T−CY transforming all arrows of S into isomorphisms. The pseudo-functor T rS(T−CX, −) is representable.

Proof. Let FS be the family of all triangular functors which map S to isomorphisms, and let S¯ denote the family of arrows which are transformed into isomorphisms by all functors from FS. Since the category CX is svelte, there exists a subset Ω of FS such that the family of all arrows of CX made invertible by functors of Ω coincides with S¯. The product of any set of quasi-cosuspended categories is a quasi-cosuspended category. In particular, the product CXΩ of targets of functors of Ω is a quasi-cosuspended FΩ category and the canonical functor CX −→ CXΩ is a triangle functor. By C3.8.2, the func- FS tor FΩ factors through a triangle localization T−CX −−−→ T−CS¯−1X. The triangle functor FS is the universal arrow representing the pseudo-functor T rS(T−CX, −).

C3.9. Triangle subcategories. A full subcategory B of the category CX is called a triangle subcategory of T−CX if it is θX-stable and has the following property: any f morphism M −→ L of B is embedded into a triangle

h g f θX(L) −→ N −→ M −→ L such that N ∈ ObB. A full triangle subcategory B of T−CX is called a thick triangle subcategory if it is h g f closed under extensions, i.e. if θX(L) −→ N −→ M −→ L is a triangle with L and N objects of B, then M belongs to B. C3.9.1. Saturated triangle subcategories. A full triangle subcategory B of a quasi-cosuspended category T−CX is called saturated if it coincides with its Karoubian envelope in T−CX; i.e. any retract of an object of B is an object of B. Evidently, every thick triangle subcategory of T−CX is saturated. It is known that the converse is true if T−CX is a triangulated category: a full triangle subcategory of a triangulated category is thick iﬀ it is saturated.

C3.10. Triangle subcategories and multiplicative systems. Let T−CX = (CX, θX, T rX) be a quasi-cosuspended k-linear category; and let B be its triangle subcat- t egory. Let Σ(B) denote the family of all arrows N −→ M of the category CX such that h t f there exists a triangle θX(L) −→ N −→ M −→ L with L ∈ ObB. Set

n Σ∞(B) = {s ∈ HomCX | θ (s) ∈ Σ(B) for some n ≥ 0}.

150 C3.10.1. Proposition. Let B be a full triangle subcategory of a quasi-cosuspended category T−CX = (CX, θX, T rX). Then the class Σ∞(B) is a multiplicative system. It is saturated iﬀ the subcategory B is saturated.

Proof. It follows from the deﬁnitions of Σ(B) and Σ∞(B) that both systems are θX-stable and contain all isomorphisms.

For a full triangle subcategory B of the quasi-cosuspended category T−CX, we set −1 CX/B = Σ(B) CX.

C3.10.2. Proposition. Let T−CX and T−CY be quasi-cosuspended categories, and F let T−CX −→ T−CY be a triangle functor. Then (a) Ker(F ) is a thick triangle subcategory of T−CX; (b) θX(ΣF ) ⊆ Σ(Ker(F )) ⊆ ΣF . In particular, ΣF = Σ(Ker(F )) if the quasi- cosuspension θX is a conservative functor. h g f Proof. (a) If θX(L) −→ N −→ M −→ L is a triangle in CX with L and N objects of Ker(F ), then the functor F maps it to the triangle 0 −→ 0 −→ F (M) −→ 0, hence F (M) = 0. (b) Let N −→t M be a morphism of Σ(F ); i.e. there exists a triangle

h t f θX(L) −→ N −→ M −→ L with L ∈ ObKer(F ). The functor F maps it to the triangle

F (t) 0 −→ F (N) −→ F (M) −→ 0 which means, precisely, that F (t) is an isomorphism. This shows that Σ(Ker(F )) ⊆ ΣF . s h g s Conversely, let M −→ L be a morphism of ΣF and θX(L) −→ N −→ M −→ L a triangle. The functor F maps it to the triangle

∼ F (h) F (g) ∼ ... −→ F θX(M) −→ F θX(L) −→ F (N) −→ F (M) −→ F (L).

n n Therefore, F θX(N) = 0 for all n ≥ 0. This shows that θX(s) ∈ Σ(Ker(F )) for n ≥ 1. C3.11. Coaisles and t-structures in a quasi-cosuspended category.

C3.11.1. Coaisles in a quasi-cosuspended category. Let T−CX = (CX, θX, T rX) be a quasi-cosuspended category. Its thick triangle subcategory U is called a coaisle if the j∗ ∗ inclusion functor U −→ CX has a left adjoint, j .

C3.11.2. Proposition [KeV1]. Let T−CX = (CX, θX, T rX) be a triangulated k- linear category (i.e. the quasi-cosuspension θX is an auto-equivalence). Then a strictly full subcategory U of CX is a coaisle iﬀ it is θX-stable and for each M ∈ ObCX, there is a triangle U U θX(M ) −→ M⊥U −→ M −→ M , (1)

151 U ⊥ where M is an object of U and M⊥U is an object of U. The triangle (1) is unique up to isomorphism.

Proof. Suppose U is a coaisle in T−CX, i.e. it is θX-stable and the inclusion functor j∗ ∗ η ∗ U −→ CX has a left adjoint, j . Fix an adjunction morphism IdCX −→ j∗j . Then we have, for any M ∈ ObCX, a triangle

∂(M) k(M) η(M) ∗ U ∗ U θXj∗j (M) = θX(M ) −−−→ M⊥U = K(M) −−−→ M −−−→ j∗j (M) = M (2)

Since, by hypothesis, j∗ is a triangle functor, its application to the triangle (2) produces a ∗ ∗ triangle in the quasi-cosuspended category T−U. Since j η is an isomorphism, j (K(M)) = ∗ 0, i.e. K(M) = M⊥U belongs to the kernel of the localization functor j . It is easy to see that Ker(j∗) coincides with ⊥U. U Conversely, suppose that for every M ∈ ObCX, there exists a triangle (1) with M ∈ ⊥ ObU and M⊥U ∈ Ob U.

j∗ C3.11.2. Cores of t-structures. The core of a t-structure U −→ CX is the subcat- ⊥ egory U ∩ θX(U). C4. Cohomological functors on suspended categories. Universal cohomological and homological functors. See preliminaries on exact and (co)suspended categories in Appendix K. Categories (suspended, cosuspended, exact) and functors of this section are k-linear for some ﬁxed commutative unital ring k.

C4.1. k-Presheaves on a k-linear Z+-category. Fix a k-linear Z+-category ∗ ΘX (CX, θX). Let Mk(X) −−−→ Mk(X) denote the continuous (i.e. having a right adjoint) θX extension of the functor CX −−−→ CX. This extension is determined uniquely up to isomorphism by the quasi-commutativity of the diagram

θX C −−−→ C X X   hXy y hX ∗ ΘX Mk(X) −−−→ Mk(X) where hX is the Yoneda embedding. ∗ Let Θ∗ be a right adjoint to ΘX. Notice that the projective objects of the category ∗ Mk(X) are direct summands of coproducts of representable presheaves. Since ΘX maps representable presheaves to representable objects and preserves arbitrary coproducts, it maps projective objects of Mk(X) to projective objects. Therefore, thanks to the fact that the category Mk(X) has enough projectives, the functor Θ∗ is exact.

C4.1.1. Note. Whenever it is convenient, we shall identify a k-linear Z+-category ∗ (CX, θX) with the equivalent to it full subcategory of the Z+-category (Mk(X), ΘX) generated by representable presheaves.

152 + C4.2. Cohomological and homological functors. Let T+CX = (CX, θX, TrX) be Φ a suspended category and (CZ , EZ ) an exact category. A functor CX −→ CZ is called a Φ cohomological functor on T+CX with values in (CZ , EZ ) (and we write T+CX −→ (CZ , EZ )), if for any triangle L −→ M −→ N −→ θX(L), the sequence

Φ(L) −→ Φ(M) −→ Φ(N) −→ Φ(θX(L)) −→ Φ(θX(M)) −→ ... (1)

f is ’exact’ and for any morphism L −→ M of CX, there exists a kernel of Φ(f) and the canonical monomorphism Ker(Φ(f)) −→ Φ(L) is an inﬂation. − Ψ Dually, if T−CX = (CX, θX, TrX) is a cosuspended category, then a functor CX −→ CZ Ψ op Ψop op is called a homological functor T−CX −→ (CZ , EZ ) if the dual functor CX −→ CZ is cohomological. In other words, for any triangle θX(N) −→ L −→ M −→ N, the sequence

... −→ Ψ(θX(M)) −→ Ψ(θX(N)) −→ Ψ(L) −→ Ψ(M) −→ Ψ(N)

f is ’exact’ and for any morphism L −→ M of CX, there exists a cokernel of Ψ(f) and the canonical epimorphism Ψ(M) −→ Cok(Ψ(f)) is a deﬂation in (CZ , EZ ). − C4.2.1. Example. Let T−CX = (CX, θX, TrX) be a k-linear cosuspended category. Then for every W ∈ ObCX, the sequence

... −→ CX(W, θX(M)) −→ CX(W, θX(L)) −→ CX(W, N) −→ CX(W, M) −→ CX(W, L) (2) is exact. This means precisely that the Yoneda embedding

hX CX −−−→ Mk(X),M 7−→ CX(−,M), is a homological functor. + Let T+CX = (CX, θX, TrX) be a suspended category. For every object V of CX and every triangle L −→ M −→ N −→ θX(L), the sequence

CX(L, V ) ←− CX(M,V ) ←− CX(N,V ) ←− CX(θX(L),V ) ←− CX(θX(M),V ) ←− ... (3) o is exact. In other words, the functor hX dual to the Yoneda embedding

op o CX −−−→ Mk(X ),M 7−→ CX(M, −), is a cohomological functor.

C4.3. Universal homological functors.

C4.3.1. The category CXa . For any k-linear category CX , let CXa denote the full subcategory of the category Mk(X) of k-presheaves on CX whose objects are k-presheaves having a left resolution formed by representable presheaves.

153 C4.3.2. Proposition. (a) The subcategory CXa is closed under extensions; i.e. CXa is an exact subcategory of the abelian category Mk(X). In particular, CXa is an additive k-linear category. (b) Suppose that the category CX is Karoubian. Let

0 −→ M 0 −→ M −→ M 00 −→ 0

0 00 be an exact sequence in Mk(X). If two of the objects M ,M,M belong to the subcategory

CXa , then the third object belongs to CXa . (b1) More generally, if CX is Karoubian and

0 −→ Mn −→ Mn−1 −→ ... −→ M2 −→ M1 −→ 0

is an exact sequence in Mk(X) with at least n−1 objects from the subcategory CXa , then the remaining object belongs to CXa . 0 00 Proof. (a) Let 0 −→ M −→ M −→ M −→ 0 be an exact sequence in Mk(X). Let P 0 −→ M 0 and P 00 −→ M 00 be projective resolutions. Then, by [Ba, I.6.7], there exists a diﬀerential on the graded object P = P 0 ⊕ P 00 such that the splitting exact sequence 0 −→ P 0 −→ P −→ P 00 −→ 0 is an exact sequence of complexes which are resolutions of the exact sequence 0 −→ M 0 −→ M −→ M 00 −→ 0. If the complexes P 0 and P 00 are formed by representable presheaves, then P is a complex of representable presheaves, hence M is an object of the subcategory CXa . (b) The assertion (b) follows from [Ba, I.6.8] and (b1) is a special case of [Ba, I.6.9]. − C4.3.3. Lemma. If T−CX = (CX, θX, TrX) is a cosuspended category, then objects of CXa are all objects M of Mk(X) such that there exists an exact sequence

M1 −→ M0 −→ M −→ 0, where M0 and M1 are representable presheaves. f e Proof. In fact, let M1 −→ M0 −→ M −→ 0 be such an exact sequence. Since M0 and ∗ d g f M1 are representable, there exists a triangle ΘX(M0) −→ M2 −→ M1 −→ M0 which gives rise to a resolution

∗ Θ (f) d g f e ∗ X ∗ ... −−−→ ΘX(M1) −−−→ ΘX(M0) −−−→ M2 −−−→ M1 −−−→ M0 −−−→ M of the object M. − C4.3.4. Proposition. Let T−CX = (CX, θX, TrX) be a cosuspended category. Then HX hX the corestriction CX −−−→ CXa of the Yoneda embedding CX −−−→ Mk(X) to the subcategory CXa is a universal homological functor in the following sense: for any exact category H (CZ , EZ ) and a homological functor T−CX −−−→ (CZ , EZ ), there exists a unique up to Ha isomorphism ’exact’ functor (CXa , EXa ) −−−→ (CZ , EZ ) such that H'Ha ◦ HX.

154 θXa The category CXa has a unique up to isomorphism Z+-category structure CXa −→ CXa such that the functor HX is a Z+-functor (CX, θX) −→ (CXa , θXa ). I Proof. (a) Fix an exact category (CZ , EZ ) with the class of deﬂations EZ . Let qZ denote the Gabriel-Quillen embedding CZ −→ CZE . Since CZE is a Grothendieck category, H in particular it is cocomplete (i.e. closed under colimits), any functor CX −→ CZ gives a rise to a quasi-commutative diagram

H C −−−→ C X Z   I hXy y qZ (1) H∗ Mk(X) −−−→ CZE

∗ ∗ in which the functor H has a right adjoint, H∗. Since the functor H preserves colimits of small diagrams (thanks to the existence of a right adjoint) and every object of the category ∗ Mk(X) is a colimit of a small diagram of representable presheaves, H is determined uniquely up to isomorphism by the quasi-commutativity of the diagram (1). H If CX −→ CZ is a homological functor T−CX −→ (CZ , EZ ), then the composition I qZ I of H and CZ −→ CZE is a homological functor, because the functor qZ is ’exact’ and homological functors are stable under the composition with ’exact’ functors. (b) The diagram (1) induces the quasi-commutative diagram

H C −−−→ C X Z   I HXy y qZ (2) ∗ Ha CXa −−−→ CZE

∗ ∗ The claim is that the functor Ha (– the restriction of the functor H to CXa ) is ’exact’. 0 00 ∗ In fact, let M −→ M −→ M be a conﬂation in CXa . Since the functor H is right exact, the sequence H∗(M 0) −→ H∗(M) −→ H∗(M 00) −→ 0 is exact. It remains to show that H∗(M 0) −→ H∗(M) is a monomorphism. 0 0 00 00 0 f 0 e 0 00 f 00 e 00 Let P1 −→ P0 −→ M −→ 0 and P1 −→ P0 −→ M −→ 0 be exact sequences in 0 0 00 0 f 0 CXa such that the objects Pi ,Pi , i = 0, 1, are representable. The morphisms P1 −→ P0 00 0 0 0 00 f 00 ∗ 0 d 0 g 0 f 0 and P1 −→ P0 can be inserted into triangles resp. ΘX(P0) −→ P2 −→ P1 −→ P0 and 00 00 00 ∗ 00 d 00 g 00 f 00 ΘX(P0 ) −→ P2 −→ P1 −→ P0 which give rise to the complexes

∗ 0 Θ (f ) d0 g0 f 0 0 ∗ 0 X ∗ 0 0 0 0 P = ... −−−→ ΘX(P1) −−−→ ΘX(P0) −−−→ P2 −−−→ P1 −−−→ P0 and

∗ 00 Θ (f ) d00 g00 f 00 00 ∗ 00 X ∗ 00 00 00 00 P = ... −−−→ ΘX(P1 ) −−−→ ΘX(P0 ) −−−→ P2 −−−→ P1 −−−→ P0

155 By (the argument of) C4.3.2(a), there is a commutative diagram

0 −−−→ P0 −−−→ P −−−→ P00 −−−→ 0    0   00 e y e y y e (3) 0 −−−→ M 0 −−−→ M −−−→ M 00 −−−→ 0

0 00 in which P0 −→e M 0, P00 −→e M 00, and P −→e M are projective resolutions and

0 −→ P0 −→ P −→ P00 −→ 0

∗ is an exact sequence of projective complexes. Since H ◦ hX is a cohomological functor, the complexes H∗(P0) and H∗(P00) are exact. Together with the exactness of the sequence

0 −→ H∗(P0) −→ H∗(P) −→ H∗(P00) −→ 0 this implies the exactness of the complex H∗(P). Now it follows from the commutative diagram

0 −−−→ H∗(P0) −−−→ H∗(P) −−−→ H∗(P00) −−−→ 0    ∗ 0  ∗   ∗ 00 H (e )y H (e) y y H (e ) 0 −−−→ H∗(M 0) −−−→ H∗(M) −−−→ H∗(M 00) −−−→ 0 that H∗(M 0) −→ H∗(M) is a monomorphism; hence the sequence

0 −→ H∗(M 0) −→ H∗(M) −→ H∗(M 00) −→ 0 is exact. Ha ∗ I (c) There is a unique up to isomorphism functor CXa −→ CZ such that Ha ' qZ ◦Ha.

The functor Ha is an ’exact’ functor (CXa , EXa ) −→ (CZ , EZ ). f e Let M be an object of CXa , and let P1 −→ P0 −→ M −→ 0 be an exact sequence with representable objects P0 and P1. Since H is a homological functor, there exists a ∗ cokernel of the morphism H(f). We set Ha(M) = Cok(H(f)). Since the functor H is f e right exact, it maps P1 −→ P0 −→ M −→ 0 to an exact sequence. Therefore, because I qZ the Gabriel-Quillen embedding (CZ , EZ ) −→ (CZE , EZE ) is an ’exact’ functor, we have an I ∗ I isomorphism qZ (Ha(M)) 'H (M). Since the functor qZ is fully faithful, it follows that the object Ha(M) is defined uniquely up to isomorphism. By a standard argument, once g the objects Ha(M) and Ha(N) are fixed, any morphism M −→ N determines uniquely a morphism Ha(M) −→ Ha(N). ∗ I The ’exactness’ of Ha follows from the isomorphism Ha ' qZ ◦Ha, because the functor ∗ I 0 00 Ha is ’exact’ (by (b) above) and the functor qZ reflects ’exactness’: if L −→ L −→ L is I 0 I I 00 a sequence in CZ such that the sequence 0 −→ qZ (L ) −→ qZ (L) −→ qZ (L ) −→ 0 is exact, then L0 −→ L −→ L00 is a conflation. ∗ I I ∗ I (d) The isomorphism Ha ' qZ ◦Ha implies that qZ ◦(Ha◦HX) 'Ha◦HX ' qZ ◦H. Since I the functor qZ is fully faithful, it follows that H'Ha ◦ HX. It follows from the definition

156 of the exact category CXa and the exactness of the functor Ha that it is determined by the isomorphism H'Ha ◦ HX uniquely up to isomorphism. ∗ ∗ ΘX θX (e) The extension Mk(X) −→ Mk(X) of the functor CX −→ CX maps representable presheaves to representable presheaves and has a right adjoint functor. In particular, ∗ ΘX is a right exact functor, and it maps an exact sequence P1 −→ P0 −→ M −→ 0 in Mk(X) with representable presheaves P1 and P0 to an exact sequence of the same type. ∗ ∗ By C4.3.3, this implies that the subcategory CXa is ΘX-stable. Therefore, ΘX induces a θXa functor CXa −→ CXa such that the diagram

θX C −−−→ C X X   HXy y HX θXa CXa −−−→ CXa quasi-commutes, i.e. HX is a Z+-functor (CX, θX) −→ (CXa , θXa ). C4.3.5. Remarks. (a) The universal property described in C4.3.4 determines the HX exact category (CXa , EXa ) and the functor CX −→ CXa uniquely up to equivalence.

(b) It follows from the deﬁnition of the category CXa that its projective objects are retracts of representable presheaves. In particular, if the category CX is Karoubian, then every projective object of the exact category CXa is isomorphic to an object of the form HX HX (M) for some M ∈ ObCX. In other words, the canonical embedding CX −→ CXa induces an equivalence between CX and the full subcategory of the category CXa generated by all projective objects of CXa . The following proposition is a cosuspended version of Theorem 2.2.1 in [Ve2].

C4.3.6. Proposition. The map which assigns to each cosuspended category T−CX = − (CX, θX, TrX) the exact category CXa is functorial in the following sense: to every triangle functor Φe = (Φ, φ) from a cosuspended category T−CX to a cosuspended category T−CY, Φa there corresponds an ’exact’ Z+-functor (CXa , EXa ) −−−→ (CYa , EYa ) which maps projectives to projectives. The functor Φa is determined uniquely up to isomorphism by the quasi-commutativity of the diagram

Φ CX −−−→ CY     HXy y HY (1) Φa CXa −−−→ CYa

Proof. (a) Since Φe = (Φ, φ) is a triangle functor and HY is a homological functor, the composition, HY ◦Φ is a homological functor. By the universal property of the homological HX functor CX −→ CXa (see C4.3.4), there exists a unique (up to isomorphism) exact functor Φea CXa −−−→ CYa such that the diagram (1) quasi-commutes. The quasi-commutativity of

157 the diagram (1) implies that Φea maps representable presheaves to representable presheaves.

Since projective objects of the categories CXa and CYa are all possible retracts (direct summands) of representable presheaves, it follows that Φea maps projectives to projectives. φ a φa a The isomorphism Φ ◦ θX −→ θY ◦ Φ induces an isomorphism Φa ◦ ΘX −→ ΘY ◦ Φa, a ∗ where ΘX is the endofunctor CXa −→ CXa induced by ΘX. So that the pair (Φa, φa) is a a a Z+-functor (CXa , ΘX) −→ (CXa , ΘX) and the diagram (1) is a diagram of Z+-functors.

Let T−CX be a cosuspended category and (CZ , EZ ) an exact category. We denote by

Ex((CXa , EXa ), (CZ , EZ )) the category whose objects are ’exact’ functors from (CXa , EXa ) to (CZ , EZ ) and morphisms are morphisms of functors. Let Hom(CX,CZ ) denote the category whose objects are functors from CX to CZ and morphisms are morphisms of functors.

HX C4.3.7. Proposition. The composition with the functor CX −−−→ CXa deﬁnes a fully faithful functor

Ex((CXa , EXa ), (CZ , EZ )) −−−→ Hom(CX,CZ ) which induces an equivalence of the category Ex((CXa , EXa ), (CZ , EZ )) with the full subcategory of Hom(CX,CZ ) generated by homological functors. Proof. The assertion is a corollary of (actually, it is equivalent to) C4.3.4. − − C4.3.8. Triangle functors. Let T−CX = (CX, θX, TrX) and T−CY = (CY, θY, TrY) be cosuspended categories, and let Φe = (Φ, φ) be a triangle functor T−CX −→ T−CY. Then we have a quasi-commutative diagram of Z+-categories and Z+-functors

HX QXa (CX, θX) −−−→ (CXa , θXa ) −−−→ (CXE , ΘXE )   a a    ∗ (1) Φ y y Φa y ΦE HY QYa (C , θ ) −−−→ (C , θ ) −−−→ (C E , Θ E ) Y Y Ya Ya Ya Ya in which QXa and QYa are Gabriel-Quillen embeddings, the functor Φa is exact, and the ∗ functor ΦE has a right adjoint, ΦE∗, which is an exact functor.

C4.4. The category CXm and abelianization of triangulated categories. Fix a − k-linear cosuspended category T−CX = (CX, θX, TrX). We denote by CXm the strictly full subcategory of the category Mk(X) of k-presheaves on CX whose objects are (isomorphic to) images of morphisms between representable presheaves. In other words, an object of

Mk(X) belongs to CXm iﬀ it is a subobject and a quotient object of some representable presheaves. An immediate consequence of this description is that the category CXm is

Karoubian. It is easy to show that the subcategory CXm is closed under ﬁnite coproducts in Mk(X); i.e. CXm is an additive subcategory of Mk(X).

Notice that CXm is a subcategory of CXa . In fact, by the deﬁnition of the subcategory e CXm , for every its object M, there exist an epimorphism M0 −→ M and a monomorphism j M −→ L0, where M0 and L0 are representable presheaves. There is a triangle d g j◦e ∗ ΘX(L0) −−−→ M1 −−−→ M0 −−−→ L0.

158 Since this triangle is an exact sequence, we have an exact sequence

g e M1 −→ M0 −→ M −→ 0

with M0,M1 representable presheaves. By C4.3.3, M is an object of CXa .

It follows that an object of CXa belongs to the subcategory CXm iﬀ it is a subobject of a representable presheaf.

∗ C4.4.1. Proposition. (a) The subcategory CXm is ΘX-stable. ∗ (b) For every morphism α of CXm , the kernel and cokernel of ΘX(α) belong to the subcategory CXm .

0 α 0 f 0 f 0 Proof. (i) Let K −→ K be morphism of CXm ; i.e. there exist M −→ L and M −→ L such that K = Im(f),K0 = Im(f 0), and presheaves M and M 0 are representable. Let 0 0 0 ∗ h g f ∗ 0 h 0 g 0 f 0 ΘX(L) −→ N −→ M −→ L and ΘX(L ) −→ N −→ M −→ L be triangles. Then there is a commutative diagram

∗ ΘX(f) h g e j Θ∗ (M) −−−→ Θ∗ (L) −−−→ N −−−→ M −−−→ K −−−→ L X  X     ∗      ΘX(ξ1) y ξe0 y y ξ2 y ξ1 y α ∗ 0 Θ (f ) h0 g0 e0 j0 ∗ 0 X ∗ 0 0 0 0 0 ΘX(M ) −−−→ ΘX(L ) −−−→ N −−−→ M −−−→ K −−−→ L (7) ξ constructed as follows. The arrow M −→1 M 0 is due to the fact that M is a projective 0 0 e 0 ξ2 0 object of Mk(X) and M −→ K is an epimorphism. Similarly, the morphism N −→ N g0 0 exists because the sequence N 0 −→ M 0 −→e K0 is exact and the object N is projective. By the property (SP2), the sequences

∗ −Θ (f) h g ∗ X ∗ ΘX(M) −−−→ ΘX(L) −−−→ N −−−→ M and ∗ 0 −Θ (f ) h0 g0 ∗ 0 X ∗ 0 0 0 ΘX(M ) −−−→ ΘX(L ) −−−→ N −−−→ M

∗ ξe0 ∗ 0 are triangles. By (SP3), there exists a morphism ΘX(L) −→ ΘX(L ) such that the diagram

∗ −ΘX(f) h g Θ∗ (M) −−−→ Θ∗ (L) −−−→ N −−−→ M X  X    ∗     ΘX(ξ1) y ξe0 y y ξ2 y ξ1 ∗ 0 −Θ (f ) h0 g0 ∗ 0 X ∗ 0 0 0 ΘX(M ) −−−→ ΘX(L ) −−−→ N −−−→ M commutes. Therefore, the diagram (7) commutes.

159 ∗ ∗ ∗ 0 (ii) Since the functor ΘX is right exact, the arrows ΘX(e) and ΘX(e ) in the commutative diagram

∗ ∗ ∗ ΘX(g) ΘX(e) ΘX(j) h Θ∗ (N) −−−→ Θ∗ (M) −−−→ Θ∗ (K) −−−→ Θ∗ (L) −−−→ N X X X X   ∗   ∗ ∗    ΘX(ξ2) y y ΘX(ξ1)ΘX(α) y ξe0 y y ξ2 ∗ 0 ∗ 0 ∗ 0 Θ (g ) Θ (e ) Θ (j ) h0 ∗ 0 X ∗ 0 X ∗ 0 X ∗ 0 0 ΘX(N ) −−−→ ΘX(M ) −−−→ ΘX(K ) −−−→ ΘX(L ) −−−→ N (8) ∗ are epimorphisms. It follows from the exactness of the rows in (7) that the arrows ΘX(j) ∗ 0 and ΘX(j ) are monomorphisms. An argument similar to that of [Ve2, 3.2.5] applied to the commutative diagram (7) ∗ shows that the kernel and cokernel of the morphism ΘX(α) belong to the subcategory CXm .

Since α is an arbitrary morphism of CXm , it follows, in particular, that the subcategory CXm ∗ is ΘX-stable; i.e. it has a natural structure of a Z+-category and the Yoneda embedding ∗ induces a Z+-functor (CX, θ) −→ (CXm , ΘX).

hX C4.4.2. Note. Since the Yoneda functor CX −→ Mk(X) takes values in CXm , the

Z+-category (CXm , ΘXm ) has enough projectives. It follows that the ’translation’ functor ΘX m ∗ CXm −−−→ CXm induced by ΘX maps projectives to projectives.

θX C4.4.3. Proposition. Suppose that the cosuspension functor CX −→ CX is a cate- − gory equivalence, i.e. T−CX = (CX, θX, TrX) is a triangulated category. Then CXm is an abelian category which coincides with CXa .

θX Proof. If the suspension functor CX −→ CX is a category equivalence, then its ex- ∗ tension ΘX is a category equivalence. In this case, it follows from C4.4.1(ii) that the subcategory CXm contains kernels and cokernels of all its morphisms, hence CXm is an abelian subcategory of Mk(X). Since every object of the category CXa is the cokernel of a morphism between representable objects, it follows that CXa ⊆ CXm . Therefore

CXa = CXm . C4.4.4. Note. Proposition C4.4.3 together with 3.2.4 and 5.2.6 recover, in particular, the ’abelianization’ theory for triangulated categories [Ve2, II.3]. C4.5. Triangulation and abelianization of cosuspended categories.

C4.5.1. Inverting endofunctors. A Z-category (CX, θX) is called strict if the endofunctor θX is an auto-morphism of the category CX. There is a standard construction which assigns to each Z+-category (CX, θX) a strict

Z-category (CXs , θXs ). Objects of the category CXs are pairs (n, M), where n ∈ Z and M ∈ ObCX. Morphisms are deﬁned by

def n−s n−t CXs ((s, M), (t, N)) = colimn≥s,tCX(θX (M), θX (N)). (1) The composition is determined by the compositions

n−r n−s n−s n−t n−r n−t CX(θX (L), θX (M)) × CX(θX (M), θX (N)) −−−→ CX(θX (L), θX (N)).

160 The functor θXs is deﬁned on objects by θXs (s, M) = (s − 1,M). It follows from (1) above that there is a natural isomorphism

∼ CXs ((s, M), (t, N)) −→ CXs (θXs (s, M), θXs (t, N)) = CXs ((s − 1,M), (t − 1,N)),

which is the action of θXs on morphisms. ΦX There is a functor CX −−−→ CXs which maps an object M of CX to the object (0,M) and a morphism M −→ N to its image in

def n n CXs ((0,M), (0,N)) = co lim CX(θX(M), θX(N)). n≥1

The morphism

ϕX (M) θXs ◦ ΦX(M) = (−1,M) −−−→ ΦX ◦ θX(M) = (0, θX(M)) is the image of the identical morphism θX(M) −→ θX(M). Let Z+ − Catk denote the category of svelte k-linear Z+-categories, and let Z − Catk denote its full subcategory generated by k-linear strict Z-categories.

C4.5.1.1. Proposition. The map which assigns to a Z+-category (CX, θX) the strict J∗ Z-category (CXs , θXs ) extends to a functor Z+ − Catk −−−→ Z − Catk which is a left J∗ adjoint to the inclusion functor Z − Catk −−−→ Z+ − Catk.

(ΦX,ϕX)

Proof. The morphisms (CX, θX) −−−→ (CXs , θXs ) deﬁned above form an adjunc- ∗ tion morphism from identical functor on Z − Catk to the composition J∗J . The second adjunction morphism is a natural isomorphism. C4.5.2. Cosuspended categories and strict triangulated categories. The construction of C4.5.1 extends to a functor from the category of cosuspended categories to the category of strict triangulated categories. Recall that a triangulated category TCX = − (CX, θX, T rX ) is strict if θX is an auto-morphism of the category CX. − C4.5.2.1. Proposition [KeV]. To any cosuspended category T−CX = (CX, θX, T rX ), there corresponds a strict triangulated category T−CXs and a triangle functor

(ΦX,ϕX ) T−CX −−−→ T−CXs such that for every triangulated category TCY, the functor

− − Trfk (T−CXs , TCY) −−−→ Trfk (T−CX, TCY) (1) of composition with (ΦX, ϕX) is an equivalence of categories. (a) If TCY is a strict triangulated category, then (1) is an isomorphism of categories. (b) If T−CX is a triangulated category, then (ΦX, ϕX) is a triangle equivalence.

161 Proof. By C4.5.1, objects of the category T−CXs are pairs (n, M), where n ∈ Z and M ∈ ObCX. The triangles are sequences

θXs (r, L) = (r − 1,L) −→ (t, N) −→ (s, M) −→ (r, L) associated to sequences

n−r w n−t v n−s u n−r θXθX (L) −→ θX (N) −→ θX (M) −→ θX (L) such that ((−1)nw, v, u) is a triangle.

Let T−Catk (resp. TrCatk) denote the category whose objects are svelte cosuspended (resp. svelte triangulated strict) k-linear categories and morphisms are triangle functors. C4.5.3. Proposition. The map which assigns to each cosuspended category the tJ∗ corresponding strict triangulated category extends to a functor T−Catk −−−→ TrCatk which is a left adjoint to the inclusion functor. Proof. See C4.5.1.1.

C4.5.4. Proposition. Let T−CX be a cosuspended k-linear category. The functor J∗ Z+ − Catk −−−→ Z − Catk maps the natural embedding CXm −→ CXa of Z+-categories to an equivalence between abelian strict Z-categories. Proof. It follows from the construction of the functor J∗ that it is compatible with tJ∗ the ’triangularization’ functor T−Catk −−−→ TrCatk of C4.5.3. The constructions of the categories CXm and CXa are also compatible with the functors triangularization functor and ∗ the functor J . By C4.4.3, the categories CXm and CXa coincide if T−CX is a triangulated category, hence the assertion. C4.6. Complements. C4.6.0. Exact categories and exact categories with enough projectives. Let

(CX , EX ) be an exact category and CXP its full subcategory generated by all objects M of CX such that there exists a deﬂation P −→ M, where P is a projective object of (CX , EX ).

It follows from (the argument of) C4.3.2 that the subcategory CXP is fully exact (i.e. it is closed under extensions). In particular, it is an exact subcategory of (CX , EX ). By construction, this exact subcategory, (CXP , EXP ), has enough projectives. Let Catex denote the bicategory of exact categories (whose 1-morphisms are ’exact’ P functors) and Catex its full subcategory generated by exact categories with enough projectives. The map which assigns to every exact category (CX , EX ) its fully exact subcategory P (CXP , EXP ) extends to a 2-functor from Catex to Catex which is left adjoint to the inclusion P functor Catex −→ Catex (in the 2-categorical sense).

C4.6.1. Costable categories in terms of complexes. Let (CX , EX ) be an exact category. Consider the full subcategory CP0X of the homotopy category H(CX ) whose d2 d1 d0 objects are acyclic complexes P = (... −→ P2 −→ P1 −→ P0 −→ M −→ 0) such that objects Pi, i ≥ 0, are projective.

162 The category CP0X has a natural Z+-action given by the ’translation’ functor θ− d2 d1 d0 which assigns to every object P = (... −→ P2 −→ P1 −→ P0 −→ M −→ 0) the object d2 d1 d0 θ−(P) = (... −→ P2 −→ P1 −→ Cok(d0) −→ 0).

C4.6.1.1. Lemma. Let (CX , EX ) be an exact category with enough projectives. Then the costable category CS−X of CX is Z+-equivalent to the category CP0X .

Proof. The equivalence is given by the functor CP0X −→ CS−X which assigns to d2 d1 d0 every object P = (... −→ P2 −→ P1 −→ P0 −→ M −→ 0) of CP0X the (image in CS−X d0 of the) cokernel of P1 −→ P0. The quasi-inverse functor assigns to each object M of CS−X

(the image in CP0X of) its projective resolution (... −→ P2 −→ P1 −→ P0 −→ M −→ 0). It follows from the deﬁnitions that both functors are compatible with the Z+-actions on the respective categories. C4.6.2. Homological dimension.

C4.6.2.1. Proposition. Let (CX , EX ) be an exact category with enough projectives, PX T−CS−X = (CS−X , θ, TrS−X ) its costable cosuspended category, and CX −→ CS−X the canonical projection. (a) The following condition on an object M of CX are equivalent: (a1) hd(M) ≤ n; n (a2) θ (PX (M)) = 0. (b) An object M of CX is projective iff its image in the costable category is zero. Proof. Consider first the case n = 0. Then the condition (a1) means that the object M is projective. The condition (a2) reads: the image of M in the costable category is zero. The implication (a) ⇒ (b) follows from the definition of the costable category. On the other hand, the image of M in the costable category is zero iff the image of the identical morphism idM is zero. The latter means that idM factors through a projective object, i.e. M is a retract of a projective object, hence it is projective. d1 d0 Suppose now that n ≥ 1. Let (... −→ P1 −→ P0 −→ M) be a projective resolution of the object M. By the definition of the cosuspension θ, there is an isomorphism n θ(PX (M)) ' PX (im(d0)). Therefore, θ (PX (M)) ' PX (im(dn−1). The homological dimension of M is less or equal to n iff im(dn−1) is a projective object, or, equivalently, PX (im(dn−1) = 0.

C4.6.2.1.1. Corollary. Let (CX , EX ) be an exact category with enough projectives. The following conditions are equivalent: (a) hd(CX , EX ) ≤ n; (b) θn = 0. In particular, hd(CX , EX ) = 0 iﬀ the costable category of (CX , EX ) is trivial. C4.6.2.2. Homological dimension of objects of a cosuspended category. Let − T−CX = (CX, θX, TrX) be a cosuspended category. We say that an object M of CX has n n−1 homological dimension n if θ (M) = 0 and θ (M) 6= 0. In particular, an object of CX is of homological dimension zero iﬀ it is zero.

163 − C4.6.2.3. Proposition. Let T−CX = (CX, θX, TrX) be a cosuspended category.

(a) The full subcategory CXhω of the category CX generated by the objects of ﬁnite homological dimension is a thick cosuspended subcategory of T−CX.

(b) The subcategory CXhω is contained in the kernel of the canonical ”triangulariza- (ΦX,ϕX ) tion” functor T−CX −−−→ T−CX(Z) (see C4.5.2.1.)

Proof. (a) Recall that a full cosuspended subcategory B of T−CX is called a thick h g f cosuspended subcategory if it is closed under extensions, i.e. if θX(L) −→ N −→ M −→ L is a triangle and L and N are objects of B, then M is an object of B too. w v u By K8.4(b), for every triangle θX(L) −→ N −→ M −→ L, the sequence of representable functors

CX(−,w) CX(−,v) CX(−,u) ... −−−→ CX(−, θX(L)) −−−→ CX(−,N) −−−→ CX(−,M) −−−→ CX(−,L) is exact. In particular, there is an exact sequence of representable functors

n n n ... −−−→ CX(−, θX(N)) −−−→ CX(−, θX(M)) −−−→ CX(−, θX(L)) −−−→ ... (1) for every positive integer n. If the objects L and N have finite homological dimension, i.e. n n θX(L)) and θX(N)) are zero objects for some n, then it follows from the exactness of the n sequence (1) that θX(M)) = 0. (b) Triangulated categories are precisely cosuspended categories whose cosuspension functor is an auto-equivalence. Therefore, every nonzero object of a triangulated category has an infinite homological dimension. C4.6.2.4. Homological dimension of a cosuspended category. Homological dimension of the cosuspended category T−CX is, by definition, the supremum of homological n dimensions of its objects. In particular, hd(CX) ≤ n for some finite n iff θX = 0. C4.6.3. The stable and costable categories of an arbitrary exact category. Let (CX , EX ) be an exact category with the class of deflations (resp. inflations) EX (resp. I qX MX ). Let CX −→ CXE be the Gabriel-Quillen embedding. Since CXE is a Grothendieck category, it has enough injectives. In particular, CXE has the stable suspended category

(CS+XE , ΘXE , TrS+XE ) with inﬁnite coproducts and products.

The composition of the Gabriel-Quillen embedding and the projection CXE −→ CS+XE gives a functor CX −→ CS+XE . We call the stable category of the exact category CX the triple (CT+X , ΘX , TrT+X ), where CT+X is the smallest ΘXE -invariant full subcategory of

CS+XE containing the image of CX ,ΘX is the endofunctor of CT+X induced by ΘXE , and

TrT+X is the class of all triangles from TrS+XE which belong to the subcategory CT+X .

One can see that (CT+X , ΘX , TrT+X ) is a full suspended subcategory of the suspended category (CT+XE , ΘXE ). If the exact category CX has enough injectives, then the suspended category (CT+X , ΘX ) is equivalent to the stable category of CX deﬁned earlier.

The costable category (CT−X , θX , TrT−X ) of the exact category CX is deﬁned dually. C4.6.4. Canonical resolutions.

164 C4.6.4.1. The resolution of a cosuspended category. Let T−CX = (CX, θX, TrX) be a cosuspended category. The universal homological functor is the full embedding the HX Z+-categories (CX, θX) −−−→ (CXa , ΘXa ) which realizes CX as a subcategory of the full sucategory of CXa generated by projective objects of (CXa , EXa ). Since the exact category

CXa has enough projectives, its costable category CS−Xa is (the underlying category of) a cosuspended category with the cosuspension functor θ2. Since the functor ΘXa maps projectives to projectives, it induces an endofunctor θ1 on the costable category CS−Xa .

It follows from the exactness of the functor ΘXa that θ1 ◦ θ2 ' θ2 ◦ θ1; i.e. CTXa is a cosuspended Z+-category. In particular, it is a Z+ × Z+-category. The canonical universal homological functor embeds the cosuspended Z+-category CS−Xa into an exact Z+ × Z+- category C(S−Xa)a , etc.. As a result of this procedure, we obtain a sequence of categories and functors P H P HX Xa X1 Xa,1 CX −−−→ CX −−−→ CX −−−→ CX −−−→ ... a 1 a,1 (1) PX H P H a,n−1 Xn Xa,n Xn+1 ... −−−→ CXn −−−→ CXa,n −−−→ CXn+1 −−−→ ... where Xa,n = (Xn)a, Xn+1 = S−Xa,n for n ≥ 0 and X0 = X. n n+1 It follows that Xn is represented by a cosuspended Z+-category (hence a Z+ - n+1 category), Xa,n is represented by an exact Z+ -category; and the universal homological functor H and the canonical projections P are n -functors. All exact categories Xn Xa,n Z+

(CXa,n , EXa,n ) have enough projectives. For every exact category (CX , EX ) with enough projectives, let ΦX denote the com- PX position of the projection CX −→ CS−X to the costable category and the universal homo-

HS−X logical functor CS−X −−−→ CS−Xa . Set Φ = H ◦ P . Then we have a sequence of functors n Xn Xa,n−1 Φ Φ Φ HX Xa Xa,1 Xa,2 CX −−−→ CX −−−→ CX −−−→ CX −−−→ ... a a,1 a,2 (2) Φ Φ Φ Φ Xa,n−2 Xa,n−1 Xa,n Xa,n+1

... −−−→ CXa,n−1 −−−→ CXa,n −−−→ CXa,n+1 −−−→ ... in which the composition of any two consecutive arrows equals to zero. The kernel of the Φ Xa,n functor CXa,n −−−→ CXa,n+1 coincides with the full subcategory of the category CXa,n generated by all its projective objects. It coincides with the Karoubian envelope in CXa,n of the image of the functor Φ . Xa,n−1 C4.6.4.2. The resolution of an exact category with enough projectives. Let (CX , EX ) be an exact category with enough projectives. Let CPX denote the full subcategory of the category CX generated by all projectives of (CX , EX ). Then we have a sequence H P H KX PX X0 Xa,0 X1 CPX −−−→ CX −−−→ CX −−−→ CX −−−→ CX −−−→ CX ... 0 a,0 1 a,1 (3) PX H P H a,n−1 Xn Xa,n Xn+1 ... −−−→ CXn −−−→ CXa,n −−−→ CXn+1 −−−→ ...

165 where X0 = S−X, i.e. CX0 is the costable category of the exact category (CX , EX ), and the rest is deﬁned as in (1) above. Again, one can ignore the intermediate cosuspended categories and obtain a complex of exact categories

Φ Φ KX ΦX Xa,0 Xa,1 CPX −−−→ CX −−−→ CX −−−→ CX −−−→ ... a,0 a,1 (4) Φ Φ Φ Φ Xa,n−1 Xa,n Xa,n+1 Xa,n+2

... −−−→ CXa,n −−−→ CXa,n+1 −−−→ CXa,n+2 −−−→ ...

C4.6.4.3. Note. If T−CX = (CX, θX, TrX) is a triangulated category (i.e. θX is n an auto-equivalence), then all the cosuspended Z+-categories T−CXn = (CXn , θXn , TrXn ) n n constructed above are triangulated Z -categories and all exact Z+-categories CXa,n are abelian Zn-categories. C5. The weak costable category of a right exact category.

C5.1. Deﬁnition. Let (CX , EX ) be a right exact category such that the category CX has an initial object, x. We denote by Pr(X, EX ) the full subcategory of CX whose objects are projectives. Let SeX denote the class of all arrows t1 in the commutative diagram

k(e0) e Ker(e0) −−−→ P −−−→ M    t   t  id 1y y 0 y M k(e) e Ker(e0) −−−→ V −−−→ M

0 where e, e are deﬂations, t0 (hence t1) are split epimorphisms, and P (hence V ) is an object of Pr(X, EX ). Let SX be the smallest saturated system containing SeX and all 0 0 −1 deﬂations P −→ P with P and P in Pr(X, EX ). We call the quotient category SX CX the weak costable category of the right exact category (CX , EX ) and denote it by CS−X .

C5.1.1. Proposition. Let (CX , EX ) be a right exact category with initial objects and w enough projectives. For any object N of the costable category, let θX (N) denote the image e in CS−X of Ker(e), where P −→ N is a deﬂation with P projective (we identify objects of w CS−X with objects of CX ). The object θX (N) is determined uniquely up to isomorphism. w The map N 7−→ θX (N) extends to a functor CS−X −→ CS−X .

0 00 Proof. Let P 0 −→e N ←−e P 00 be deﬂations with P 0 and P 00 projective objects. Since (CX , EX ) has enough projectives, there exists (by the argument C5.3.1(a)) a commutative diagram 0 t0 P −−−→ P 0   00  0 t0 y y e e00 P 00 −−−→ N

166 whose arrows are deﬂations and the object P is projective. Therefore, we have a commutative diagram

k0 e00 Ker(e0) −−−→ P 0 −−−→ N x x x 0   0  t1  t0  idN k e00 Ker(e) −−−→ P −−−→ N    00  00  t1 y y t0 y idN k00 e00 Ker(e00) −−−→ P 00 −−−→ N

0 00 Since t0 and t0 are deﬂations to projective objects, they are split epimorphisms. Therefore, 0 00 t1 and t1 are split epimorphisms, i.e. they belong to SeX (cf. 2.5). f 0 Consider a diagram N −→ L ←−e M, where e0 is a deﬂation. Then we have a commutative diagram

k(σ) σ Ker(σ) −−−→ P −−−→ N       t1y y t0 y idN k(e) e Ker(e) −−−→ N −−−→ N (1)       f1y y f0 y f k(e0) e0 Ker(e0) −−−→ M −−−→ L in which the right lower square is cartesian, the morphism f1 is uniquely determined by the choice of f0 (hence both f0 and f1 are determined by f uniquely up to isomorphism), t0 is a deﬂation, and t1 is (a deﬂation) uniquely determined by t0. Applying the localization q∗ SX CX −→ CS−X , we obtain morphisms

∗ ∗ q (t1) q (f1) SX SX θw (N) −→∼ q∗ (Ker(σ)) −−−→ q∗ (Ker(e)) −−−→ q∗ (Ker(e0)). (2) X SX SX SX

The only choice in this construction is that of the deﬂation P −→t0 N. If P 0 −→s0 N is another choice, then there exists a commutative square

00 s0 P 00 −−−→ P   00  t0 y y t0 s0 P 0 −−−→ N

00 00 00 whose arrows are deﬂations and the object P is a projective. Therefore, t0 and s0 are

167 split deﬂations, and we have a commutative diagram

k(σ0) σ0 Ker(σ0) −−−→ P 0 −−−→ N x x x 00  00  s1   s0  idN k(σ00) σ00 Ker(σ00) −−−→ P 00 −−−→ N (3)    00  00  t1 y y t0 y idN k(σ) σ Ker(σ) −−−→ P −−−→ N whose vertical arrows belong to SX , i.e. their images in the costable category are isomorphisms. This implies that the composition θw (N) −→ q∗ (Ker(e0)) of morphisms of (2) X SX does not depend on the choice of the deflation P −→t0 N. Taking M projective, we obtain a θ (f) w X w w morphism θX (N) −−−→ θX (L) which is uniquely defined once the choice of objects θX (N) w and θX (L) is fixed.

C5.2. The weak cosuspension functor. Let (CX , EX ) be a right exact category with enough projectives and initial objects. Let CS−X its cosuspended category. The w θX functor CS−X −−−→ CS−X deﬁned in C5.1.1 is called the weak cosuspension functor.

Notice that the weak costable category CS−X of (CX , EX ) has initial objects. If the category CX is pointed, then CS−X is pointed and the image in CS−X of each projective object of (CX , EX ) is a zero object.

C5.2.1. Note. It follows from C6.7.1 that if the category CX is additive, then the w weak costable category CS−X with the weak cosuspension functor θX is equivalent to the costable category CT−X with the cosuspension functor θX . C5.3. Right exact categories of modules over monads and their weak costable categories. Suppose that CX is a category with initial objects and such that spl spl the class EX of split epimorphisms of CX is stable under base change; so that (CX , EX ) is a right exact category. Let F = (F, µ) be a monad on CX . Set CX = F − mod. f∗ ∗ Let CX −→ CX be the forgetful functor, f its canonical left adjoint, and ε the stan- ∗ dard adjunction morphism f f∗ −→ IdCX . We denote by EX the right exact structure spl on CX induced by EX via the forgetful functor f∗. By 5.5, (CX, EX) is a right exact category with enough projectives: all modules of the form (F (L), µ(L)),L ∈ ObCX , are projective objects of (CX, EX), and every module M = (M, ξ) has a canonical deflation ε(M) ∗ f f∗(M) −−−→ M. ∗ ε We denote by ΩF the kernel of the adjunction morphism f f∗ −→ IdCX and call it the functor of Kählerdifferentials.

C5.3.1. Standard triangles. Let M = (M, ξM) and L = (L, ξL) be F-modules f∗(e) and M −→e L a deﬂation (i.e. the epimorphism M −−−→ L splits). Then we have a

168 commutative diagram

k (L) ε(L) F ∗ ΩF (L) −−−→ f f∗(L) −−−→ L       (4) ∂ y y t0 y idL k e Ker(e) −−−→ M −−−→ L which contains (and deﬁnes) the standard triangle

∂ k e ΩF (L) −−−→ Ker(e) −−−→ M −−−→ L (5) corresponding to the deﬂation M −→e L.

The image of (5) in the weak costable category CS−X is a standard triangle of CS−X .

C5.4. Example: right exact categories of unital algebras. Let CX be the category Algk of associative unital k-algebras. The category CX has an initial object – the k-algebra k, and the associated pointed category CXk is the category of augmented k-algebras. C5.4.1. The functor of Kählerdifferentials. Kählerdifferentials appear when ∗ f∗ f we have a pair of adjoint functors CX −→ CY −→ CX . Presently, the role of the category f∗ CY is played by the category of k-modules. The forgetful functor Algk −→ k − mod has a canonical left adjoint f ∗ which assigns to every k-module M the tensor algebra L ⊗n Tk(M) = n≥0 M . Therefore, the class of all split k-module epimorphisms induces via f∗ a structure EX of a right exact category on CX = Algk. In this case, the tensor algebra ∗ f (M) = Tk(M) is a projective object of (CX , EX ) for every k-module M; and for every k-algebra A, the adjunction morphism

ε(A) ∗ f f∗(A) = Tk(f∗(A)) −−−→ A,

(determined by the k-algebra structure and the multiplication f∗(A)⊗k f∗(A) −→ f∗(A) in A) is a canonical projective deflation. By definition, the functor Ωk of Kählerdifferentials assigns to each k-algebra A the kernel of the adjunction morphism ε(A), which coincides + + with the augmented k-algebra k ⊕ Ωk (A), where Ωk (A) is the kernel K(ε(A)) of the algebra morphism ε(A) in the usual sense (i.e. in the category of non-unital algebras).

C5.4.2. The functor of non-additive K¨ahlerdiﬀerentials. The category Algk has small products and kernels of pairs of arrows A ⇒ B, hence it has limits of arbitrary f∗ small diagrams. As any functor having a left adjoint, the forgetful functor Algk −→ k−mod preserves limits. In particular, f∗ preserves pull-backs and, therefore, kernel pairs of algebra ϕ morphisms. Therefore, each k-algebra morphism A −→ B has a canonical kernel pair p −→1 A ×B A −→ A. Using the fact that A ×B A is computed as the pull-back of k-modules, we p2 can represent A ×B A as the k-module f∗(A) ⊕ K(f∗(ϕ)) with the multiplication induced by the isomorphism

∼ f∗(A) ⊕ Ker(f∗(ϕ)) −−−→ f∗(A) ×f∗(B) f∗(A), x ⊕ y 7−→ (x, x + y).

169 That is the multiplication is given by the formula (a ⊕ b)(a0 ⊕ b0) = aa0 ⊕ (ab0 + ba0 + bb0). We denote this algebra by A#K(ϕ). ∗ ε Applying this to the adjunction arrow f f∗ −→ IdCX , we obtain a canonical isomor- ∗ + phism between the functor Ωek of non-additive K¨ahlerdiﬀerentials and f f∗#Ωk , where ε(A) + Ωk (A) is the kernel of the algebra morphism Tk(f∗(A)) −−−→ A in the category of non- unital k-algebras (cf. C5.4.1). Thus, for every k-algebra A, we have a commutative diagram similar to the one in the additive case: k(ε) ε + ∼ −−−→ k ⊕ Ωk (A) −→ Ωk(A) −−−→ Tk(f∗(A)) −−−→ A 0   k       ejky jky y id y id (6) λ 1 ε + ∼ −−−→ Tk(f∗(A))#Ωk (A) −→ Ωek(A) −−−→ Tk(f∗(A)) −−−→ A λ2 Here 0k = 0k(A) is the ’zero’ morphism – the composition of the augmentation morphism Ωk(A) −→ k and the k-algebra structure k −→ Tk(f∗(A)). The morphism ejk (hence jk) becomes an isomorphism in the costable category. s C5.4.3. Another canonical right exact structure. Let EX denote the class s of all strict epimorphisms of k-algebras. The class EX is stable under base change, i.e. (CX , EX ) is a right exact category. For every projective k-module V , the tensor algebra s f∗ Tk(V ) is a projective object of (CX , EX ), because the forgetful functor Algk −→ k − mod is exact (hence it maps strict epimorphisms to epimorphisms of k-modules). By 5.3.1, its ∗ s left adjoint f maps projectives of k − mod to projectives of (CX , EX ). That is for every s projective k-module V the tensor algebra Tk(V ) of V is a projective object of (CX , EX ). ∗ ε Since the adjunction arrow f f∗ −→ IdCX is a strict epimorphism and k−mod has enough s projectives, the right exact category (CX , EX ) has enough projectives: for any k-algebra e A, there exists a strict k-algebra epimorphism Tk(V ) −→ A for some projective k-module V . By 2.2.1, the kernel Ker(e) coincides with the augmented k-algebra k ⊕ K(e), where K(e) is the kernel of e in the usual sense – a two-sided ideal in Tk(V ).

f∗ C5.4.4. Remarks. (a) The forgetful functor Algk −→ k − mod is conservative and preserves cokernels of pairs of arrows. Therefore, by Beck’s Theorem, there is a canonical equivalence (in this case, an isomorphism) between the category Algk and the category ∗ F − mod of modules over the monad F = (f∗f , µ) = (Tk(−), µ) associated with the pair ∗ ∗ ε of adjoint functors f∗, f and the adjunction morphism f f∗ −→ IdAlgk . op (b) Consider the category Affk = Algk of affine (noncommutative) k-schemes. Right exact structures on Algk define left exact structures on Affk and vice versa. Inflations in Affk corresponding strict epimorphisms of algebras are precisely closed immersions of (noncommutative) affine schemes. (c) The example C5.4 is generalized to algebras in an additive monoidal category. C5.5. The left exact category of comodules over a comonad and its weak stable category. Fix a comonad G = (G, δ) on a category CX with final objects and split spl monomorphisms stable under cobase change; i.e. (CX , IX ) is a left exact category.

170 C5.5.1. The suspension functor. Let G+ denote the functor CY −→ CY which assigns to every G-comodule M = (M, ν) the cokernel of the adjunction morphism

ν ∗ M −−−→ g∗g (M) = (G(M), δ(M))

(see 5.5.2(2)). The functor G+ is a canonical suspension functor on the category CY = G − comod = (Y\G) − comod which induces a suspension functor on the stable category S+CY of the exact category (CY, EY).

φ 0 C5.5.2. Lemma. A morphism M −→ M of CY becomes a trivial morphism in the stable category T+CY iﬀ it factors through an adjunction arrow (3); i.e. there exists a commutative diagram φ M −−−→ M0 ν &% h ∗ g∗g (M)

h ∗ 0 for some morphism g∗g (M) = (G(M), δ(M)) −−−→ M .

φ 0 Proof. By definition of the stable category, the image of an arrow M −→ M of CY in the stable category T+CY is trivial iff it factors through an EY-injective object N . By 5.5.3, ∗ φ 0 the adjunction arrow N −→ g∗g (N ) splits. Therefore, the arrow M −→ M becomes trivial in the stable category iff it factors through a morphism M −→ g∗(N) for an object ∗ N of CX . Every such arrow factors through the adjunction morphism M −→ g∗g (M); hence the assertion. j e C5.5.3. Standard triangles. For any conflation L −→ M −→ N in CY = G − comod, the standard triangle

j e d L −→ M −→ N −→ G+(L) is deﬁned via a commutative diagram

j e L −−−→ M −−−→ N       idLy y γ y d (1) ηγ (L) λγ (L) L −−−→ G(L) −−−→ G+(L)

∗ λg where G = g∗g and G −→ G+ is the canonical deﬂation. The morphism γ in (1) exists d by the EY-injectivity of G(L). The morphism N −→ G+(L) is uniquely determined by the choice of γ (because e is an epimorphism). The image of d in the stable category T+CY does not depend on the choice of γ.

f C5.6. Frobenious morphisms of ’spaces’ and Frobenious monads. Let Y −→ X be a continuous morphism of ’spaces’ with an inverse image functor f ∗ and a direct image

171 functor f∗. We say that f is a Frobenious morphism if there exists an auto-equivalence Ψ ∗ of the category CX such that the composition f ◦ Ψ is a right adjoint to f∗. It is clear that every isomorphism is a Frobenious morphism and the composition of Frobenious morphisms is a Frobenious morphism. f It follows that every Frobenious morphism Y −→ X with a conservative direct image functor is aﬃne. Therefore, the category CY can be identiﬁed with the category F − mod of modules over the monad F = (F, µ) on a category CX associated with the pair of ∗ adjoint functors f , f∗. Conversely, we call a monad F on the category CX a Frobenious f∗ monad if the forgetful functor F −mod −−−→ CX is a direct image functor of a Frobenious Ψ morphism; i.e. there exists an equivalence CX −→ CX such that the functor

f ∗◦Ψ CX −−−→ F − mod, V 7−→ (F (Ψ(V )), µ(Ψ(V )), is a right adjoint to the forgetful functor f∗. In particular, the monad F is continuous.

C5.6.1. Proposition. Let F be a Frobenious monad on a category CX such that spl (CX , EX ) is a right exact category. Then the right exact category (CX, EX), where CX is spl the categor F − mod of F-modules and EX is a right exact structure induced by EX , is a Frobenious category.

∗ Proof. Let f∗ denote the forgetful functor F − mod −→ CX and f its canonical left ! ∗ adjoint. Let Ψ be a functor CX −→ CX such that the composition f = f ◦ Ψ is a right adjoint to f∗. Then every injective object of the category F − mod is a retract of an object ∗ of the form f (Ψ(V )) for some V ∈ ObCX . On the other hand, every projective object of ∗ F − mod is a retract of an object of the form f (L) for some L ∈ ObCX . Therefore, every injective F-module is projective. If the functor Ψ is an auto-equivalence, then f ∗ ' f ! ◦Ψ∗, ∗ ! where Ψ is a quasi-inverse to Ψ. That the functor f ◦Ψ is a left adjoint to f∗. By duality, it follows from the argument above that every projective object of F − mod is injective. C5.7. The costable category associated with an augmented monad. Let F =

(F, µ) be an augmented monad on a k-linear additive category CX ; i.e. F = IdCX ⊕ F+. The category F − mod of F-modules is isomorphic to the category F+ − mod1 of F+- actions. Recall that the objects of F+ − mod1 are pairs (M, ξ), where M ∈ ObCX and ξ is a morphism F+(M) −→ M satisfying associativity condition with respect to multiplication 2 µ+ F+ −→ F+, i.e. ξ ◦ µ+(M) = ξ ◦ F+(ξ). Morphisms are deﬁned naturally. Notice that the monad F is continuous (i.e. the functor F has a right adjoint) iﬀ the functor F+ has a right adjoint. ∗ ∗ It follows that ΩF −→ f f∗ factors through the subfunctor F+ of f f∗ corresponding + to the subsemimonad (F+, µ ) of F. The full subcategory TF+ of F − mod generated by all F-modules M such that ΩF (M) −→ F+(M) is an isomorphism (i.e. the action of F+ on M is zero) is isomorphic to the category CX .

C5.7.1. Inﬁnitesimal neighborhoods. Let T (n) denote the n-th inﬁnitesimal F+ neighborhood of TF+ , n ≥ 1. It is the full subcategory of F − mod generated by modules

172 + n ξn M = (M, ξ) such that the n-th iteration F+(M) −→ M of the action of F+ on M is zero. In particular, T (1) = T . F+ F+ + Since ξn is an F-module morphism for any n ≥ 1, an F-module M = (M, ξ) is an object of T (n) iﬀ F ·n ,→ Ω , where F ·n is the image of the iterated multiplication F+ + F + + n µn ·n F+ −→ F+. One can see that F+ is a two-sided ideal in the monad F. If the quotient ·n functor F/F+ is well deﬁned (which is the case if cokernels of morphisms exist in CX ), ·n then there is a unique monad structure µn on the quotient F/F+ such that the quotient ·n ·n ·n morphism F −→ F/F+ is a monad morphism from F to F/F+ = (F/F+ , µn) and the category T (n) is equivalent to the category F/F ·n-modules. Clearly, F/F ·n is an F+ + + ·n ·n augmented monad: F/F+ ' IdCX ⊕ F+/F+ . ·(n−1) ·n ·n It follows from the preceeding discussion that F /F ,→ Ω ·n ,→ F /F . + + F/F+ + + ·2 In particular, Ω ·2 = F+/F . F/F+ +

C5.7.2. Free actions. Let CX be a k-linear category with the exact structure spl E ; and let L be a k-linear endofunctor on CX . Consider the category L − act whose objects are pairs (M, ξ), where M ∈ ObCX and ξ is a morphism L(M) −→ M. Morphisms between actions are defined in a standard way. We endow L − act with the exact structure f∗ induced by the forgetful functor L − act −→ CX . If CX has countable coproducts and the functor L preserves countable coproducts, then the category L − act is isomorphic to T(L) − mod, where T(L) = (T (L), µ) is a free monad generated by the endofunctor n L; i.e. T (L) = ⊕n≥0L and µ is the multiplication defined by the identical morphisms Ln ◦ Lm −→ Ln+m, n, m ≥ 0. The category CX is isomorphic to the full subcategory TL of L−act generated by zero (n) actions. The n-th infinitesimal neighborhood of TL is the full subcategory TL of L − act ξ generated by all actions (M, ξ) such that the n-th iteration Ln(M) −→n M of the action (n+1) ξ is zero. The category TL is equivalent to the category TL,n − mod of modules over M m the monad TL,n = (TL,n, µn), where TL,n = L and the multiplication defined by 0≤m≤n morphisms Lk ◦ Lm −→ Lk+m, 0 ≤ k, m ≤ n, which are identical if k + m < n and zero otherwise. n + def M m It follows from C5.7.1 that L ,→ ΩTL,n ,→ TL,n = L . 1≤m≤n

In particular, ΩTL,2 = L. Here L denotes the functor L−act −→ L−act which assigns to an object (M, ξ) the object (L(M), L(ξ)) and acts on morphisms accordingly. C5.7.2.1. Projectives and injectives of an inﬁnitesimal neighborhood. Pro- (n+1) jective objects of the category TL = TL,n − mod are retracts of relatively free objects. The latter are TL,n-modules of the form TL,n(V ),V ∈ ObCX . M m Suppose that L has a right adjoint functor, L∗. Then the functor TL,n = L 0≤m≤n ! M m has a right adjoint equal to TL,n = L∗ ; that is TL,n is a continuous monad. 0≤m≤n

173 Therefore, by G1.4, the injective objects of TL,n − mod are retracts of TL,n-modules of the ! ! form TL,n(V ) = (TL,n(V ), γn(V )),V ∈ ObCX .

C5.7.2.2. Proposition. Suppose that L is an autoequivalence of the category CX . (n+1) Then TL = TL,n − mod is a Frobenius category.

Proof. It suﬃces to show that TL,n is a Frobenious monad. An adjunction arrow n L ◦ L∗ −→ IdCX induces a canonical morphism from TL,n(L∗ (V )) to the injective object ! TL,n(V ). If L is an autoequivalence, then this canonical morphism is an isomorphism.

C5.7.3. Example. Let CX be the product of Z copies of a k-linear category CY ; i.e. objects of CX are sequences M = (Mi| i ∈ Z) of objects of CY . Let L be the translation functor: L(M)i = Mi−1. Objects of the category L − act of L-actions are arbitrary dn+1 dn dn−1 (n) sequences of arrows (... −→ Mn+1 −→ Mn −→ ...). Objects of the subcategory TL are sequences such that the composition of any n consecutive arrows is zero. In particular, (2) (1) TL coincides with the category of complexes on CY and its subcategory TL = TL is the (n) category of complexes with zero diﬀerential. By C5.7.2.2, TL is a Frobenious category for every n. Therefore, its costable category is triangulated. Notice that the costable category (2) of TL coincides with the homotopy category of unbounded complexes.

174 Appendix K. Exact categories and their (co)stable categories.

K1. Exact categories. We follow here the approach of B. Keller [Ke1]. For the convenience of applications, we consider mostly k-linear categories and k-linear functors, where k is a commutative associative unital ring. For a k-linear category CX , we denote op by Mk(X) the category of k-linear functors CX −→ k − mod and will call it the category of k-presheaves on CX .

K1.1. Definition. Let CX be a k-linear category and EX a class of pairs of morphisms j e j e L −→ M −→ N of CX such that the sequence 0 −→ L −→ M −→ N −→ 0 is exact (i.e. j is a kernel of e and e a cokernel of j). The elements of EX are called conflations. The morphism e (resp. j) of a conflation L −→j M −→e N is called a deflation (resp. inflation). The pair (CX , EX ) is called an exact category if EX is closed under isomorphisms and the following conditions hold. (Ex0) id0 is a deflation. (Ex1) The composition of two deflations is a deflation. f (Ex2) For every diagram M 0 −→ M ←−e L, where e is a deflation, there is a cartesian square e0 L0 −−−→ M 0   0   f y y f e L −−−→ M where e0 is a deflation. f (Ex2op) For every diagram M 0 ←− M −→j L, where j is an inflation, there is a cocartesian square j0 L0 ←−−− M 0 x x 0   f   f j L ←−−− M where j0 is an inflation. For an exact category (CX , EX ), we denote by EX the class of all deflations and by MX the class of all inflations of (CX , EX ). K1.2. Remarks.

K1.2.1. Applying (Ex2) to (Ex0), we obtain that idM is a deflation for every M ∈ ObCX . Thus, axioms (Ex0), (Ex1), (Ex2) mean simply that the class EX of deflations forms a right multiplicative system, or, what is the same, a pretopology on CX . The invariance of EX under isomorphisms implies that all isomorphisms of CX are deflations. The fact that all arrows of EX are strict epimorphisms means precisely that the pretopology EX on CX is subcanonical, i.e. every representable presheaf of sets on CX is a sheaf on (CX , EX ). Thus, one can start from a class EX of arrows of CX which forms a subcanonical pretopology (equivalently, it is a right multiplicative system formed by strict

175 epimorphisms) and define MX as kernels of arrows of EX . The only remaining requirement op is the axiom (Ex ) – the invariance of the class MX of inflations under a cobase change. This shows, in particular, that the first three axioms make sense in any category and the last axiom, (Ex2op), makes sense in any pointed category. The fact that all identical morphisms are deflations implies that arrows 0 −→ M op are inflations for all objects M of CX . Applying the axiom (Ex2 ) to arbitrary pair of inflations L ←− 0 −→ M, we obtain the existence of coproducts of any two objects; i.e. the category CX is additive. K1.2.2. Quillen’s original definition of an exact category contains some additional axioms. B. Keller showed that they follow from the axioms (Ex0) – (Ex2) and (Ex2op) (cf. [Ke1, Appendix A]). Moreover, he observes (in [Ke1, A.2]) that the axiom (Ex2) follows from (Ex2op) and a weaker version of (Ex2): f (Ex2’) For every diagram M 0 −→ M ←−e L, where e is a deflation, there is a commutative square e0 L0 −−−→ M 0   0   f y y f e L −−−→ M where e0 is a deflation. Quillen’s description of exact categories is self-dual which implies self-duality of Keller’s op op axioms: if (CX , EX ) is an exact category, then (CX , EX ) is an exact category too. K1.2.3. In the axioms (Ex2) and (Ex2op), the conditions ”there exists a cartesian (resp. cocartesian) square” can be replaced by ”for any cartesian (resp. cocartesian) square”. This implies that for any family {Ei | i ∈ J} of exact category structures on an \ additive category CX , the intersection EJ = Ei is a structure of an exact category. i∈J K2. Examples of exact categories.

K2.1. The smallest exact structure. For any additive k-linear category CX , let spl j e spl EX denote the class of all split sequences L −→ M −→ N. Then the pair (CX , EX ) is an spl exact category. Notice that EX is the smallest exact structure on CX . K2.2. The category of complexes. Let C(A) be the category of complexes of an • • additive k-linear category A. Conﬂations are diagrams L• −→j M • −→e N • such that the jn en diagram Ln −→ M n −→ N n is split for every n ∈ Z. K2.3. Quasi-abelian categories. A quasi-abelian k-linear category is an additive k-linear category CX with kernels and cokernels and such that every pullback of a strict epimorphism is a strict epimorphism, and every pushout of a strict monomorphism is a strict monomorphism. It follows from deﬁnitions that the pair (CX , Es), where Es is the class of all short exact sequences in CX , is an exact category. Every abelian k-linear category is quasi-abelian.

176 K2.4. Filtered objects. Let (CX , EX ) be an exact k-linear category. Objects of the ﬁltered category F(CX , EX ) are sequences of inﬂations

jn M = (... −→ Mn −→ Mn+1 −→ ...) such that Mn = 0 for n 0 and jm are isomorphisms for m 0. Morphisms of filtered objects are defined in a natural way (componentwise). Conflations are sequences of two morphisms whose components belong to EX . If (CX , EX ) is a quasi-abelian category, then the filtered category F(CX , EX ) is quasi- abelian too.

K2.5. The category of Banach spaces. Let CX be the category of complex Banach spaces. A sequence of morphisms L −→j M −→e N is a conflation if it is an exact sequence of complex vector spaces. Thus defined exact category of Banach spaces is quasi-abelian. In fact, a morphism of Banach spaces is a strict epimorphism iff it is an epimorphism of vector spaces.

K2.6. Categories of functors. Let (CX , EX ) be an exact k-linear category and CZ a category. The category Hom(CZ ,CX ) of functors from CZ to CX is an exact category: a sequence F 0 −→j F −→e F 00 of functor morphisms is a conﬂation if

j(M) e(M) F 0(M) −−−→ F (M) −−−→ F 00(M) is a conﬂation for every object M of CZ .

K3. ’Exact’ functors. Let (CX , EX ) and (CY , EY ) be exact k-linear categories. F A k-linear functor CX −→ CY is called ’exact’ if it maps conﬂations to conﬂations. We denote by ExCatk the category whose objects are exact k-linear categories and morphisms ’exact’ k-linear functors.

K3.1. Example. Let CX and CY be additive k-linear categories. Every k-linear F spl F spl functor CX −→ CY is an ’exact’ functor (CX , EX ) −→ (CY , EY ) (see K2.1). The map spl which assigns to an additive k-linear category CX the exact category (CX , EX ) and to a k-linear functor the corresponding ’exact’ functor is a full embedding of the category Addk of additive k-linear categories and k-linear functors to the category ExCatk of exact k-linear categories and ’exact’ k-linear functors. This embedding is a left adjoint to the forgetful functor ExCatk −→ Addk.

K3.2. Example: ’exact’ functors from a quasi-abelian category. Let (CX , EX ) and (CY , EY ) be exact k-linear categories. If (CX , EX ) is quasi-abelian, then a k-linear F functor CX −→ CY is an ’exact’ functor from (CX , EX ) to (CY , EY ) iﬀ it preserves ﬁnite limits and colimits. In other words, ’exact’ functors in this case are precisely exact functors.

K4. Right and left ’exact’ functors. Let (CX , EX ) and (CY , EY ) be exact k-linear F categories. A k-linear functor CX −→ CY is called right ’exact’ if it maps deﬂations to

177 e deflations and for any deflation M −→ N of (CX , EX ), the functor F maps the canonical p1 −−−→ e diagram M ×N M −−−→ M −−−→ N to an exact diagram. p2 Left ’exact’ functors (CX , EX ) −→ (CY , EY ) are defined dually. op K4.1. Remark. Taking as CY the abelian category k − mod with the canonical exact structure, one can see that right ’exact’ functors from (CX , EX ) to (CY , EY ) are, precisely, sheaves of k-modules on the pretopology (CX , EX ). They also can be viewed as op def op op left ’exact’ functors from (CX , EX ) = (CX , EX ) to k − mod.

K4.2. Proposition. Let (CX , EX ) and (CY , EY ) be exact k-linear categories and F CX −→ CY a k-linear functor which maps deflations to deflations and for every deflation e M −→ N of (CX , EX ) the canonical morphism

F (M ×N M) −−−→ F (M) ×F (N) F (M) is a deﬂation of (CY , EY ). Then F is right ’exact’.

Proof. The condition that the pretopology (CY , EY ) is subcanonical means precisely −→ that for every deflation K −→ L, the diagram K ×L K −→ K −→ L is exact. Since the functor F maps deflations to deflations, the diagram

−−−→ F (M) ×F (N) F (M) −−−→ F (M) −−−→ F (N) is exact for every deﬂation M −→ N of (CX , EX ). Since by hypothesis, the canonical morphism F (M ×N M) −−−→ F (M) ×F (N) F (M) is exact, this implies the exactness of the diagram

−−−→ F (M ×N M) −−−→ F (M) −−−→ F (N).

That is F is a right ’exact’ functor.

K4.3. Proposition. Let (CX , EX ) and (CY , EY ) be exact k-linear categories and ϕ∗ (CX , EX ) −−−→ (CY , EY ) a k-linear functor. The following conditions are equivalent: (a) ϕ∗ is right ’exact’; (b) for any conﬂation L −→j M −→e N, the sequence

ϕ∗(j) ϕ∗(e) ϕ∗(L) −−−→ ϕ∗(M) −−−→ ϕ∗(N) −−−→ 0 (1) is ’exact’. ∗ ∗ (c) the functor ϕ maps deﬂations to deﬂations and ϕe∗ : F 7−→ F ◦ ϕ maps sheaves of k-modules to sheaves of k-modules.

178 Proof. (a) ⇔ (b). For any conﬂation L −→j M −→e N, we have a commutative diagram ∼ −−−→ Ker(p2) −−−→ L  0    k2 y y j p1 −−−→ e M ×N M −−−→ M −−−→ N p2 which induces a commutative diagram

∼ Ker(p2) −−−→ L     k2 y y j (2) p1−p2 e M ×N M −−−→ M −−−→ N

Since e ◦ p1 = e ◦ p2, the morphism p1 − p2 is the composition j ◦ γ for a uniquely deﬁned morphism γ. It follows from the diagram (2) and the monomorphness of j that γ is a ∗ split epimorphism. In particular, γ ∈ EX . Since γ is a split epimorphism, ϕ (γ) has this property. Therefore the sequence (1) is exact iﬀ the sequence

ϕ∗(jγ) ϕ∗(e) ∗ ∗ ∗ ϕ (M ×N M) −−−→ ϕ (M) −−−→ ϕ (N) −−−→ 0 (3) is exact. The exactness of the sequence (3) is equivalent to the exactness of the diagram

∗ ϕ (p1) ϕ∗(e) ∗ −−−→ ∗ ∗ ϕ (M ×N M) −−−→ ϕ (M) −−−→ ϕ (N). ∗ ϕ (p2)

(b) ⇒ (c). (i) The equivalence of (a) and (b) applied to presheaves of k-modules (i.e. op CY is the category k − mod with the canonical exact structure), gives a description of sheaves of k-modules on the pretopology (CX , EX ): a presheaf F of k-modules is a sheaf iff for any conflation L −→ M −→ N, the sequence 0 −→ F(N) −→ F(M) −→ F(L) is exact. (ii) It follows that the sequence 0 −→ F(N) −→ F(M) −→ F(Le) is exact for any ’exact’ sequence Le −→ M −→ N −→ 0 and any sheaf of k-modules F. In fact, the sequence Le −→ M −→ N −→ 0 being exact means that the morphism Le −→ M is the composition of a deflation Le −→ L and an inflation L −→ M such that L −→ M −→ N is a conflation. By (i) above, the latter implies that the sequence 0 −→ F(N) −→ F(M) −→ F(L) is exact and the arrow F(L) −→ F(Le) is a monomorphism for any sheaf F of k-modules. Therefore the sequence 0 −→ F(N) −→ F(M) −→ F(Le) is exact. ϕ∗ (iii) Let now F be a sheaf of k-modules on (CY , EY ). If (CX , EX ) −−−→ (CY , EY ) is a right ’exact’ functor and L −→ M −→ N is a conflation in (CX , EX ), then the sequence ϕ∗(L) −→ ϕ∗(M) −→ ϕ∗(N) −→ 0 is ’exact’, hence the sequence

0 −→ Fϕ∗(N) −→ Fϕ∗(M) −→ Fϕ∗(L) (4)

179 ∗ is exact. By (ii), this means that Fϕ is a sheaf on (CX , EX ). (c) ⇒ (b). It follows from (ii) above that the condition (c) means precisely that the diagram (4) is exact for any conﬂation L −→ M −→ N and any sheaf of k-modules F. In particular, since every representable presheaf is a sheaf, (4) is exact for every representable presheaf, or, equivalently, the diagram

ϕ∗(L) −→ ϕ∗(M) −→ ϕ∗(N) −→ 0 is exact.

K4.4. Corollary. Let (CX , EX ) and (CY , EY ) be exact k-linear categories. ϕ∗ (a) A functor CX −→ CY is ’exact’ iﬀ it is both left and right ’exact’. (b) A presheaf F of k-modules on (CX , EX ) is a sheaf iﬀ the sequence

0 −→ F (N) −→ F (M) −→ F (L) is exact for any conflation L −→ M −→ N in (CX , EX ). Proof. (a) By K4.3, the functor ϕ∗ is both right and left ’exact’ iff for any conflation L −→ M −→ N in (CX , EX ), the sequences

ϕ∗(L) −→ ϕ∗(M) −→ ϕ∗(N) −→ 0 and 0 −→ ϕ∗(L) −→ ϕ∗(M) −→ ϕ∗(N) are exact, i.e. ϕ∗(L) −→ ϕ∗(M) −→ ϕ∗(N) is a conﬂation. (b) The assertion is proved in the argument of K4.3. It is, also, a formal consequence of K4.1 and K4.3. K4.5. Remark: right ’exact’ functors between pretopologies. The assertion K4.3(c) suggests the following

K4.5.1. Deﬁnition. Let (CX , TX ) and (CY , TY ) be pretopologies. We call a functor F CX −→ CY a right ’exact’ functor from (CX , TX ) to (CY , TY ) if it maps elements of covers to elements of covers and induces a functor between the categories of sheaves. K5. Fully exact subcategories of exact categories. Gabriel-Quillen embedding. Let (CX , EX ) be an exact category. A full subcategory B of CX is called a fully exact subcategory of (CX , EX ) if it is closed under extensions; i.e. if objects L and N in a conﬂation L −→ M −→ N belong to B, then M is an object of B too.

K5.1. Proposition. Let (CX , EX ) be an exact category and B a fully exact subcategory of CX . Then EX induces on B a structure of an exact category. Proof. The condition ’B is closed under extensions’ means that for any conflation L −→j M −→e N such that L and N are objects of B, the object M is isomorphic to an j e object of B. Let L −→ M −→ N be a conflation of EB (i.e. a conflation of EX which

180 f is a diagram in B), and let N 0 −→ N be an arbitrary morphism of B. Then we have a commutative diagram j0 e0 L0 −−−→ M 0 −−−→ N 0       g y y y f (1) j e L −−−→ M −−−→ N whose rows are conflations and the right square is cartesian. It is not difficult to see that the left vertical arrow g in (1) is an isomorphism. Since the subcategory B is closed under extensions, this implies that M 0 is isomorphic to an object of B. Hence, the class of deflations EB is invariant under base change (axiom (Ex2)). Thus, EB has properties (Ex0), (Ex1), (Ex2). The remaining property, (Ex2op), follows from the fact that everything here is selfdual.

A category CX is called svelte if it is equivalent to a small category.

K5.2. Theorem (Gabriel-Quillen embedding). Let (CX , EX ) be a svelte exact k- linear category. Then there exist a Grothendieck k-linear category CY and an ’exact’ fully faithful k-linear functor (CX , EX ) −→ (CY , EY ) which induces an equivalence between (CX , EX ) and a full subcategory of CY closed under extensions.

Proof. Notice that the category Mk(X) of k-presheaves on CX is a Grothendieck category (it has a generator because the category CX is svelte and the category Mk(X) has inﬁnite coproducts). Let CXE denote the category of sheaves of k-modules on the presite

(CX , EX ). By K4.1, CXE is a full subcategory of Mk(X) whose objects are left ’exact’ op op functors (CX , EX ) −→ k − mod (or right ’exact’ functors from (CX , EX ) to k − mod ; see K4.4(b)). The inclusion functor CXE −→ Mk(X) has a left adjoint – the sheafification functor Mk(X) −→ CXE , which is exact; i.e. CXE is (equivalent to) a quotient category of the Grothendieck category Mk(X) by a Serre subcategory, SE. Therefore, CXE is a Grothendieck category itself. Since the pretopology EX on CX is subcanonical (see K1.2.1), the Yoneda embedding induces an equivalence between the exact category (CX , EX ) and a full subcategory of CXE . It remains to show that this full subcategory is closed under ∗ jX extensions and the embedding CX −→ CXE is an ’exact’ functor which reflects conflations. (i) The Yoneda embedding CX −→ Mk(X) is a left exact functor, and the sheafifi- ∗ jX cation functor Mk(X) −→ CXE is exact. Therefore, their composition CX −→ CXE is a ∗ left exact functor; in particular, it is left ’exact’. The claim is that the functor jX is right ∗ ’exact’; i.e. jX maps every deflation to an epimorphism of the category CXE . −→ In fact, let M −→ N be a deflation and M ×N M −→ M −→ N the associated exact diagram. The Yoneda embedding maps this diagram to the diagram

Mc × Mc −→ Mc −→ N,b (2) Nb −→ where Mc = CX (−,M). For any presheaf F of k-modules, the functor Mk(X)(−, F) maps the diagram (2) to the diagram isomorphic to −→ F(N) −→ F(M) −→ F(M ×N M).

181 which is exact if F is a sheaf on (CX , EX ). This shows that for every sheaf F, the functor

CXE (−, F) maps the diagram (2) to an exact diagram. Therefore the diagram (2) viewed as a diagram in the category of sheaves, is exact. γ (ii) Let N ∈ ObCX and F −→ Nb a morphism of sheaves on (CX , EX ). Regarding γ as a presheaf morphism, we represent it as the composition of the presheaf epimorphism F −→ Im(γ) and the embedding Im(γ) ,→ Nb. It follows from the exactness of the sheafification functor that γ is a sheaf epimorphism iff the sheafification functor maps the embedding Im(γ) ,→ Nb to an isomorphism; i.e. Im(γ) ,→ Nb is a refinement of N in the topology associated with the pretopology EX . The latter means that there exists a 0 0 0 e 0 0 be deflation M −→ N such that the image of be is contained in Im(γ), i.e. Mc −→ Nb is v0 the composition of a morphism Mc0 −→ Im(γ) and the embedding Im(γ) ,→ Nb. Since 0 representable functors are projective objects in Mk(X), the morphism v factors through the presheaf epimorphism F −→ Im(γ). Thus, we obtain a commutative diagram

e0 M 0 −−−→b N c b   (3) v y y id γ F −−−→ Nb

0 Suppose that the kernel of γ is representable by an object L, and let L0 −→j M 0 be 0 the kernel of M 0 −→e N. Then the diagram (3) extends to a commutative diagram

j0 e0 0 −−−→ L0 −−−→b M 0 −−−→b N −−−→ 0 b c b    ub y y v y id (4) λ γ 0 −−−→ Lb −−−→ F −−−→ Nb −−−→ 0

0 (iii) Applying (Ex2op) to the innﬂation L0 −→j M 0 and the arrow L ←−u L0, we obtain a commutative diagram

j0 e0 L0 −−−→ M 0 −−−→ N      0  u y cocart y u y id (5) j e00 L −−−→ M −−−→ N whose left square is cocartesian and both rows are conﬂations (cf. C5.3.3). (iv) The Yoneda functor assigns to the diagram (5) the commutative diagram

j0 e0 0 −−−→ L0 −−−→b M 0 −−−→b N −−−→ 0 b c b   0  ub y y ub y id (b5) bj be 0 −−−→ Lb −−−→ Mc −−−→ Nb −−−→ 0

182 whose rows are (by (i) and (iii) above) exact sequences in the category of sheaves. The latter implies that the left square of the diagram (b5) is cocartesian. In fact, let j0 L0 −−−→b M 0 b c   ub y y ν ej Lb −−−→ G be a cocartesian square. Applying the argument of (iii) above, we obtain the commutative diagram j0 e0 0 −−−→ L0 −−−→b M 0 −−−→b N −−−→ 0 b c b    ub y y ν y id ej ee 0 −−−→ Lb −−−→ G −−−→ Nb −−−→ 0 g with exact rows. Therefore, the canonical morphism G −→ Mc gives rise to the commutative diagram j e 0 −−−→ L −−−→e G −−−→e N −−−→ 0 b  b    id y y g y id (6) bj be 0 −−−→ Lb −−−→ Mc −−−→ Nb −−−→ 0 g of sheaves on (CX , EX ) with exact rows, which shows that G −→ Mc is an isomorphism. (v) The commutative diagrams (4) and (b5) give rise to the commutative diagram of sheaves j e 0 −−−→ L −−−→b M −−−→b N −−−→ 0 b c b    id y y t y id (7) λ γ 0 −−−→ Lb −−−→ F −−−→ Nb −−−→ 0 t with exact rows, which implies that Mc −→ F is an isomorphism. (vi) By (iii) above, L −→j M −→e N is a conflation. Therefore, the isomorphism (7) ∗ ei be shows also that the functor jX reflects conflations: if 0 −→ Lb −→ Mc −→ Nb −→ 0 is an i e exact sequence of sheaves on (CX , EX ), then L −→ M −→ N is a conflation.

∗ jX K5.2.1. Note. The canonical embedding CX −→ CXE of (the argument of) K5.2 is called the Gabriel-Quillen embedding. K5.3. Sheafification functor and effaceable presheaves. Recall a standard ∗ tX construction of a sheafification functor Mk(X) −→ CXE . Let HX denote the functor Mk(X) −→ Mk(X) which assigns to every presheaf F of k-modules the presheaf HX (F ) defined by −→ HX (F )(N) = colim(Ker(F (M)−→F (M ×N M))) (8)

183 where colimit is taken by the diagram EX (N) of deflations M −→ N. The morphisms τ (N) −→ F F (N) −→ Ker(F (M)−→F (M ×N M) determine a morphism F (N) −−−→ HX (F )(N) for τF every N ∈ ObCX which is functorial in N; i.e. it defines a functor morphism F −→ HX (F ). τ The function F 7−→ τF is a functor morphism IdMk(X) −→ HX (F ). e A presheaf F on (CX , EX ) is a monopresheaf if for every deflation M −→ N, the F (e) morphism F (N) −→ F (M) is a monomorphism. There are the following facts: (a) A presheaf of k-modules F is a monopresheaf (resp. a sheaf) iff the canonical τF morphism F −→ HX (F ) is a monomorphism (resp. an isomorphism). (b) The functor HX maps every presheaf of k-modules F to a monopresheaf and every monopresheaf to a sheaf. 2 It follows that the functor HX maps presheaves to sheaves and its corestriction to the ∗ subcategory CXE of sheaves of k-modules is isomorphic to the sheafification functor tX . 2 ∗ Or, what is the same, HX ' tX∗tX . ∗ Another consequence of (a) and (b) is that the kernel of tX coincides with the kernel of the functor HX . It follows from the formula (8) that a presheaf F belongs to the kernel of HX iff it is effaceable. The latter means that for every pair (N, ξ), where N ∈ ObCX and ξ is an element of F (N), there exists a deflation M −→e N such that F (e)(ξ) = 0. ξe Equivalently, for any object N of CX and any morphism Nb −→ F , there exists a deflation e e M −→ N such that the composition of Mc −→b Nb and ξe equals to zero. ∗ tX Thus, objects of the kernel SEX of the sheafification functor Mk(X) −→ CXE are ∗ precisely effaceable presheaves. Since the functor jX is a flat localization, SEX is a Serre subcategory of the category Mk(X), and the category of sheaves CXE is equivalent to the quotient category Mk(X)/SEX .

K5.4. Proposition. Let (CX , EX ) and (CY , EY ) be svelte exact k-linear categories ϕ∗ and (CX , EX ) −−−→ (CY , EY ) a right ’exact’ k-linear functor. Then there exists a functor ∗ ϕe CXE −−−→ CYE such that the diagram

ϕ∗ C −−−→ C X Y ∗   ∗ jX y y jY ∗ ϕe CXE −−−→ CYE

j∗ j∗ quasi commutes, i.e. ϕ∗j∗ ' j∗ ϕ∗. Here C −→X C and C −→Y C are Gabriel- e X Y X XE Y YE ∗ Quillen embeddings. The functor ϕe is deﬁned uniquely up to isomorphism and has a right adjoint, ϕe∗. Proof. (i) If the functor ϕ∗ is right ’exact’, then the functor

∗ Mk(Y ) −−−→ Mk(X), F 7−→ F ◦ ϕ ,

184 maps sheaves on the pretopology (CY , EY ) to sheaves on (CX , EX ); in particular, it induces ϕe∗ a functor CYE −−−→ CXE . In fact, for any arrow M −→ N of EX , consider the decomposition

∗ Y ∗ Y ∗ −−−→ ∗ ∗ ϕ (M M) −−−→ ϕ (M) ϕ (M) −−−→ ϕ (M) −−−→ ϕ (N) (1) N ϕ∗(N) of the diagram ∗ Y −−−→ ∗ ∗ ϕ (M M) −−−→ ϕ (M) −−−→ ϕ (N) (2) N

Since the right and the left arrows of the diagram (1) belong to EY , for any sheaf F on (CY , EY ) the diagram

∗ ∗ −−−→ ∗ Y ∗ Fϕ (N) −−−→ Fϕ (M) −−−→ F ϕ (M) ϕ (M) ϕ∗(N) is exact and the morphism Y Y Fϕ∗(M) ϕ∗(M) −−−→ Fϕ∗(M M) ϕ∗(N) N is a monomorphism. Therefore, the diagram

∗ ∗ −−−→ ∗ Y Fϕ (N) −−−→ Fϕ (M) −−−→ Fϕ (M M) N

∗ is exact. This shows that Fϕ is a sheaf on the pretopology (CX , EX ). ∗ ∗ ∗ ∗ ∗ (ii) The functor ϕe∗ has a left adjoint, ϕe . It follows that ϕe jX ' jY ϕ . ∗ ∗ ϕe K5.4.1. Note. Even if the functor ϕ is ’exact’, the functor CXE −→ CYE need not to be (left) exact. For instance, let (CX , EX ) (resp. (CY , EY )) be the exact category of ∗ projective A-modules (resp. B-modules) of finite type, and ϕ the functor M 7−→ B ⊗A M corresponding to an algebra morphism A −→ B. Then the category CXE is naturally B⊗ ∗ A identified with A − mod and the functor ϕe with A − mod −−−→ B − mod. Therefore, the ∗ functor ϕe is exact iff the algebra morphism A −→ B turns B into a flat right A-module. K5.5. The Gabriel-Quillen embedding and the smallest abelianization of an exact category. Fix an exact k-linear category (CX , EX ). Consider the category j∗ AX,E whose objects are fully faithful exact k-linear functors (CX , EX ) −→ (CY , EY ) such ∗ that CY is an abelian k-linear category with the canonical exact structure, j reflects exact sequences, and CX is closed under extensions in CY . A morphism between two such ∗ ∗ j k ∗ embeddings, (CX , EX ) −→ (CY , EY ) and (CX , EX ) −→ (CZ , EZ ), is a pair (g , α), where ∗ ∗ ∗ ∼ ∗ g is a functor CY −→ CZ and α a functor isomorphism g j −→ k . The composition is

185 ∗ c j deﬁned naturally. Let AX,E denote the subcategory of AX,E whose objects are (CX , EX ) −→ ∗ (CY , EY ) such that the category CY has small coproducts and morphisms are pairs (g , α) such that the functor g∗ has a right adjoint.

K5.5.1. Proposition. Let (CX , EX ) be an exact k-linear category. ∗ jX (a) The Gabriel-Quillen embedding (CX , EX ) −→ (CXE , EXE ), is a ﬁnal object of the c category AX,E . Here EXE is the canonical exact structure of the abelian category CXE , (b) The category AX,E has an initial object.

Proof. (a) The assertion follows from K5.3.

(b) Let CXa(E) be the smallest full abelian subcategory of the category CXE containing

(the image of) CX . Its objects are kernels and cokernels (taken in CXE ) of (pairs of) arrows ∗ ja in CX . The embedding CX −→ CXa(E) is an initial object of the category AX,E .

spl K5.5.2. Example. If E = EX = {split exact sequences}, then CXE coincides with the category Mk(X) of presheaves of k-modules on CX , hence CXa(E) is the smallest abelian subcategory of Mk(X) containing the image of CX .

K6. The Karoubian envelope.

K6.1. Proposition (Karoubi). Let CX be an additive k-linear category.

(a) There exists a Karoubian additive k-linear category CXK and a fully faithful k- ∗ KX linear functor CX −→ CXK such that any k-linear functor from CX to any Karoubian ∗ KX k-linear category factors uniquely up to a natural isomorphism through CX −→ CXK . ∗ (b) Every object of CXK is a direct summand of an object in KX (CX ).

Proof. Objects of the category CXK are pairs (M, p), where M is an object of the p 2 category CX and M −→ M is an idempotent endomorphism, i.e. p = p. Morphisms f (M, p) −→ (M 0, p0) are morphisms M −→ M 0 such that fp = f = p0f. The composition f g gf of (M, p) −→ (M 0, p0) and (M 0, p0) −→ (M 00, p00) is (M, p) −→ (M 00, p00). It follows from p this deﬁnition that (M, p) −→ (M, p) is the identical morphism. 0 0 0 0 The category CXK is additive with (M, p)⊕(M , p ) = (M ⊕M , p⊕p ), and an object ∗ (M, p) is a direct summand of KX (M) = (M, idM ), because

' (M, p) ⊕ (M, idM − p) = (M ⊕ M, p ⊕ (idM − p)) −→ (M, idM ).

id id Here the isomorphism corresponds to the identical morphisms M −→M M ←−M M. ∗ KX The functor CX −→ CXK assigns to every object M of CX the object (M, idM ). It is F fully faithful. A k-linear functor CX −→ CZ to a Karoubian category CZ gives rise to a FK functor CXK −→ CZ which assigns to every object (M, p) of the category CXK the kernel ∗ of F (idM − p). It follows that FK ◦ KX ' F . In particular, for every additive functor

186 F FK CX −→ CY , there exists a natural functor CXK −→ CXK such that the diagram F C −−−→ C X Y ∗   ∗ KX y y KY FK CXK −−−→ CXK quasi-commutes. The map F 7−→ FK deﬁnes a (pseudo) functor from the category of additive categories to the category of Karoubian categories which is a left adjoint to the inclusion functor. This implies, in particular, the universal property of the correspondence

CX 7−→ CXK .

The category CXK in K6.1 is called the Karoubian envelope of the category CX .

K6.2. Proposition. Let (CX , EX ) be an exact category. The Karoubian envelope

CXK has a structure of an exact category, EK , whose conflations are direct summands of conflations of E. ∗ jX Proof. Consider the Gabriel-Quillen embedding CX −→ CXE . The category CXE ∗ is abelian, hence Karoubian. It follows from K6.1 that the functor jX factors through ∗ KX CX −→ CXK , i.e. there exists a canonical morphism CXK −→ CXE which induces an equivalence between the category CXK and the full subcategory of CXE whose objects ∗ are all direct summands of objects of jX (CX ) (see the argument of K6.1). Since the ∗ subcategory jX (CX ) is closed under extensions in CXE , the image of CXK in CXE has the same property. In fact, let 0 −→ L −→ M −→ N −→ 0 be an exact sequence in CXE 0 0 ∗ 0 such that L ⊕ L and N ⊕ N are isomorphic to objects of jX (CX ) for some objects L 0 ∗ and N of CXE . Since the subcategory jX (CX ) is closed under extensions in CXE and the sequence 0 −→ L⊕L0 −→ M ⊕L0 ⊕N 0 −→ N ⊕N 0 −→ 0 is exact, the object M ⊕L0 ⊕N 0 ∗ is isomorphic to an object of jX (CX ). This shows that M is a direct summand of an object ∗ of jX (CX ) and that any exact sequence 0 −→ L −→ M −→ N −→ 0 in CXE whose objects belong to the image of CXK is a direct summand of an image of an image of a sequence in E. The assertion follows now from K5.1. K7. Injective and projective objects of an exact category. An object M of a k-linear exact category (CX , EX ) is projective if CX (M, −) is an ’exact’ functor from (CX , EX ) to k − mod. Injective objects are defined dually – they correspond to projective op op objects of the dual exact category (CX , EX ). Let P(X, EX ) denote the full subcategory of the category CX generated by projective objects of (CX , EX ). It follows that any deflation N −→ P such that P is a projective object splits. In particular, the subcategory P(X, EX ) is closed under extensions in (CX , EX ); i.e. P(X, EX ) is a fully exact subcategory of (CX , EX ). The exact structure induced on P(X, EX ) is the smallest one: it consists of split conflations. Similarly, the full subcategory I(X, EX ) of CX generated by injective objects is a fully exact subcategory of (CX , EX ).

K7.1. Proposition. Let (CX , EX ) and (CY , EY ) be exact k-linear categories and f∗ CX −→ CY a k-linear functor having a right adjoint. If f∗ is an ’exact’ functor from (CX , EX ) to (CY , EY ), then its right adjoint maps injective objects to injective objects.

187 Dually, if f∗ has a left adjoint functor, then the latter maps EY -projective objects to EX -projective objects.

f ! Proof. Let CY −→ CX be a right adjoint to the functor f∗. For any conﬂation E ∈ EX and any EY -injective objects M, the sequence CY (f∗(E),M) is exact, because, by ! hypothesis, f∗ is an ’exact’ functor. Therefore, the sequence CX (E, f (M)) is exact, i.e. ! f (M) is an EX -injective object. K7.1.1. Remark. Notice by passing that a right (or left) adjoint to a k-linear functor f ! is a k-linear functor. In fact, let CY −→ CX be a right adjoint to a k-linear functor f∗. Then we have a commutative diagram of bifunctors

f ! ! ! CY (−, −) −−−→ CX (f (−), f (−))     ·ε y y f∗ ε· ! ! ! CY (f∗f (−), −) ←−−− CY (f∗f (−), f∗f (−)) whose right vertical arrow is a k-module morphism because the functor f∗ is k-linear. Its left vertical (resp. the lower horizontal) arrow is a k-module morphism because it is the composition from the left (resp. from the right) with the adjunction morphism ! ε f∗f −→ IdCY . The composition of the right vertical and lower horizontal arrow is an adjunction isomorphism. Since it is a k-module isomorphism, its inverse is a k-module isomorphism too. Therefore, the upper horizontal arrow is a k-module morphism, which proves that f ! is a k-linear functor. K7.2. Exact categories with enough projectives or/and injectives. An exact category (CX , EX ) has enough projectives (resp. enough injectives) if for every object M of CX , there exists a deﬂation P −→ M (resp. an inﬂation M −→ P ), where P is a projective (resp. injective) object of (CX , EX ).

f∗ K7.2.1. Proposition. Let (CX , EX ) −→ (CY , EY ) be an ’exact’ functor which re- f ! ﬂects inﬂations and has a right adjoint, CY −→ CX (resp. a left adjoint). Suppose that EY consists of split sequences. Then (CX , EX ) has enough injective (resp. projective) objects.

Proof. If EY consists of split sequences, then every object of CY is injective (and ! projective). Therefore, by K7.1, every object f (M),M ∈ ObCY , is EX -injective. For η(M) ! every M ∈ ObCY , the adjunciton arrow M −−−→ f f∗(M) belongs, by hypothesis, to the class MX of inflations of (CX , EX ), because f∗(η(M)) is a split monomorphism. K8. Suspended and cosuspended categories. Suspended categories were introduced in [KeV]. In a sequel, we shall mostly use their dual version – cosuspended categories. They are defined as follows. − K8.1. Definitions. A cosuspended k-linear category is a triple (CX, θX, T rX ), where − CX is an additive k-linear category, θX a k-linear functor CX −→ CX, and T rX is a class of sequences of the form w v u θX(L) −→ N −→ M −→ L (1)

188 called triangles and satisfying the following axioms: (SP0) Every sequence of the form (1) isomorphic to a triangle is a triangle.

idM (SP1) For every M ∈ ObCX, the sequence 0 −→ M −→ M −→ 0 is a triangle. w v u (SP2) If θX(L) −→ N −→ M −→ L is a triangle, then

−θX(u) w v θX(M) −−−→ θX(L) −−−→ N −−−→ M is a triangle. 0 0 0 w v u 0 w 0 v 0 u 0 (SP3) Given triangles θX(L) −→ N −→ M −→ L and θX(L ) −→ N −→ M −→ L β and morphisms L −→α L0 and M −→ M 0 such that the square

u L ←−−− M     α y y β u0 L0 ←−−− M 0

γ commutes, there exists a morphism N −→ N 0 such that the diagram

u v w L ←−−− M ←−−− N ←−−− θX(L)         α y y β y γ y θX(α) u0 v0 w0 0 0 0 0 L ←−−− M ←−−− N ←−−− θX(L ) commutes. (SP4) For every pair of morphisms M −→u L and M 0 −→x M, there exists a commutative diagram

u v w L ←−−− M ←−−− N ←−−− θX(L) x x x x     id   x  y  id u0 v0 w0 0 0 L ←−−− M ←−−− N ←−−− θX(L) x x x    s   t  θX(u) id r Mf ←−−− Mf ←−−− θX(M) x x   r   θX(v) θX(M) ←−−− θX(N) whose two upper rows and two central columns are triangles. K8.2. Suspended categories. A suspended k-linear category is deﬁned dually; + i.e. it is a triple T+CX = (CX, θX, T rX ), where CX is an additive k-linear category, θX a + k-linear functor CX −→ CX, and T rX is a class of sequences of the form

u v w L −→ M −→ N −→ θX(L) (2)

189 such that the dual data is a cosuspended category. K8.3. Triangulated categories and (co)suspended categories. A suspended + − category T+CX = (CX, θX, T rX ) (resp. a cosuspended category T−CX = (CX, θX, T rX )) is a triangulated category iﬀ the translation functor θX is an auto-equivalence. K8.4. Properties of cosuspended and suspended categories. The following − properties of a cosuspended category T−CX = (CX, θX, T rX )) follow directly from the axioms: u (a) Every morphism M −→ L of CX can be embedded into a triangle

w v u θX(L) −→ N −→ M −→ L.

w v u (b) For every triangle θX(L) −→ N −→ M −→ L, the sequence of representable functors

CX(−,w) CX(−,v) CX(−,u) ... −−−→ CX(−, θX(L)) −−−→ CX(−,N) −−−→ CX(−,M) −−−→ CX(−,L) (3) is exact. In particular, the compositions u ◦ v, v ◦ w, w ◦ θX(u) are zero morphisms. (c) If the rows of the commutative diagram

u v w L ←−−− M ←−−− N ←−−− θX(L)         α y y β y γ y θX(α) u0 v0 w0 0 0 0 0 L ←−−− M ←−−− N ←−−− θX(L ) are triangles and the two left vertical arrows, α and β, are isomorphisms, then γ is an isomorphism too (see the axiom K8.1 (SP3)). (d) Direct sum of triangles is a triangle. w v u (e) If θX(L) −→ N −→ M −→ L, is a triangle, then the sequence

w v θX(L) −→ N −→ M −→ 0 is split exact iﬀ u = 0. (f) For an arbitrary choice of triangles starting with u, x and xu in the diagram K8.1 (SP4), there are morphisms y and t such that the second central column is a triangle and the diagram commutes. − If T−CX = (CX, θX, T rX )) is a triangulated category, i.e. the translation functor θX is an auto-equivalence, then, in addition, we have the following properties: w v u (g) A diagram θX(L) −→ N −→ M −→ L, is a triangle if (by (SP2), iﬀ)

−θX(w) w v θX(M) −−−→ θX(L) −−−→ N −−−→ M is a triangle.

190 0 0 0 w v u 0 w 0 v 0 u 0 (h) Given triangles θX(L) −→ N −→ M −→ L and θX(L ) −→ N −→ M −→ L β γ and morphisms M −→ M 0 and N −→ N 0 such that the square

v N −−−→ M     γ y y β v0 N 0 −−−→ M 0 commutes, there exists a morphism L −→α L0 such that the diagram

u v w L ←−−− M ←−−− N ←−−− θX(L)         α y y β y γ y θX(α) u0 v0 w0 0 0 0 0 L ←−−− M ←−−− N ←−−− θX(L ) commutes. w v u (i) For every triangle θX(L) −→ N −→ M −→ L, the sequence of corepresentable functors

C (w,−) C (v,−) C (u,−) X X X o ... ←−−− CX(θX(L), −) ←−−− CX(N, −) ←−−− CX(M, −) ←−−− CX(L, −) (3 ) is exact. − − K8.5. Triangle functors. Let T−CX = (CX, θX, T rX ) and T−CY = (CY, θY, T rY) be cosuspended k-linear categories. A triangle k-linear functor from T−CX to T−CY is a pair (F, φ), where F is a k-linear functor CX −→ CY and φ is a functor morphism w v u θY ◦ F −→ F ◦ θX such that for every triangle θX(L) −→ N −→ M −→ L of T−CX, the sequence

F (w)φ(L) F (v) F (u) θY(F (L)) −−−−−−−→ F (N) −−−−−−−→ F (M) −−−−−−−→ F (L) is a triangle of T−CY. It follows from this condition and the property K8.4(b) (applied to the case M = 0) that φ is invertible. The composition of triangle functors is deﬁned naturally: (G, ψ) ◦ (F, φ) = (G ◦ F, Gφ ◦ ψF ). 0 0 If (F, φ) and (F , φ ) are triangle functors from T−CX to T−CY. A morphism from (F, φ) to (F 0, φ0) is given by a functor morphism F −→λ F 0 such that the diagram

φ θY ◦ F −−−→ F ◦ θX     θYλ y y λθX φ0 0 0 θY ◦ F −−−→ F ◦ θX commutes. The composition is the compsition of the functor morphisms.

191 − Altogether gives the deﬁnition of a large bicategory Trfk formed by cosuspended k- linear categories, triangle k-linear functors as 1-morphisms and morphisms between them as 2-morphisms. Restricting to svelte cosuspended categories, we obtain the bicategory − Trk . − We denote by Trfk (resp. by Trk) the full subbicategory of Trfk whose objects are triangulated (resp. svelte triangulated) categories. Finally, dualizing (i.e. inverting all arrows in the constructions above), we obtain the + large bicategory Trfk of suspended categories and triangular functors and its subbicategory + Trk whose objects are svelte suspended categories. Thus, we have a diagram of natural full embeddings + − Trf ←−−− Trfk −−−→ Trf xk x xk       + − Trk ←−−− Trk −−−→ Trk

(F,φ) K8.6. Triangle equivalences. A triangle k-linear functor T−CX −−−→ T−CY is (G.ψ) called a triangle equivalence if there exists a triangle functor T−CY −−−→ T−CX such that the compositions (F, φ) ◦ (G, ψ) and (G, ψ) ◦ (F, φ) are isomorphic to respective identical triangle functors. It follows from K7.1.1 that the quasi-inverse triangle functor (G, ψ) is k-linear. K8.6.1. Lemma [Ke1]. A triangle k-linear functor (F, φ) is a triangle equivalence iﬀ F is an equivalence of the underlying categories.

K9. Stable and costable categories of an exact category. Let CX be a k-linear category and B its full subcategory. The class JB of all arrows of CX which factor through some objects of B is an ideal in HomCX . We denote by B\CX , or by CB\X the category having same objects as CX ; its morphisms are classes of morphisms of CX modulo the ideal JB, that is two morphisms with the same source and target are equivalent if their diﬀerence belongs to the ideal JB. We are particularly interested in this construction when (CX , EX ) is an exact k-linear category and B is the fully exact subcategory of CX generated by EX -projective or EX - injective objects of (CX , EX ). In the ﬁrst case, we denote the category B\CX by CS−X and will call it the costable category of (CX , EX ). In the second case, the notation is CS+X and the name of this category is the stable category of (CX , EX ).

K9.1. Example. Let CX be an additive k-linear category endowed with the smallest spl exact structure EX (cf. K2.1). Then the correponding costable category is trivial: all its objects are isomorphic to zero. K9.2. Exact categories with enough projectives and their costable categories. Let (CX , EX ) be an exact k-linear category with enough projectives; i.e. for each object M of CX , there exists a deﬂation P −→ M, where P is a projective object. Then the costable category CS−X of (CX , EX ) has a natural structure of a cosuspended k-linear category deﬁned as follows. The endofunctor θS−X assigns to an object M the (image

192 in CS−X of) the kernel of a deﬂation P −→ M, where P is a projective object. For f any morphism L −→ M, the morphism θS−X (f) is the image of the morphism h in the commutative diagram

j e θS X (L) −−−→ PL −−−→ L −       h y y g y f j0 e0 θS−X (M) −−−→ PM −−−→ M

A standard argument shows that objects θS−X (L) are determined uniquely up to isomorphism and the morphism θS−X (f) is uniquely determined by the choice of the objects θS−X (L) and θS−X (M). j e With each conﬂation N −→ M −→ L of (CX , EX ), it is associated a sequence

∂ ¯j ¯e θS−X (L) −−−→ N −−−→ M −−−→ L called a standard triangle and determined by a commutative diagram

ej ee θS X (L) −−−→ PL −−−→ L −       ∂e y y g y idL j e N −−−→ M −−−→ L

The morphism g here exists thanks to the projectivity of PL. The connecting morphism ∂ θS−X (L) −−−→ N is, by deﬁnition, the image of ∂e. 0 0 0 0 ∂ 0 j 0 e 0 Triangles are deﬁned as sequences of the form θS−X (L ) −→ N −→ M −→ L which are isomorphic to a standard triangle.

K9.2.1. Proposition ([KeV]). For any exact k-linear category (CX , EX ) with enough projectives, the triple T−CS−X = (CS−X , θS−X , TrS−X ) constructed above is a cosuspended k-linear category.

If (CX , EX ) is an exact category with enough injectives, then the dual construction provides a structure of a suspended category on the stable category CS+X of (CX , EX ). K9.2.2. The case of Frobenius categories. Recall that an exact category (CX , EX ) is called a Frobenius category, if it has enough injectives and projectives and projectives coincide with injectives.

K9.2.1. Proposition. If (CX , EX ) is a Frobenius category, then its costable cosuspended category T−CS−X and (therefore) the stable suspended category T+CS+X are triangulated, and are triangular equivalent one to another.

Proof. It is easy to check that if (CX , EX ) is a Frobenius category, then the translation functor θS−X is an auto-equivalence of the category CS−X . The rest follows from this fact. Details are left to the reader.

193 K9.3. Proposition. Let (CX , EX ) and (CY , EY ) be exact k-linear categories with f ∗ enough projectives. Every ’exact’ k-linear functor (CX , EX ) −−−→ (CY , EY ) which maps ∗ T−f projectives to projectives induces a triangle k-linear functor T−CS−X −−−→ T−CS−Y between the corresponding costable cosuspended categories. Proof. The argument is left to the reader.

K9.3.1. Corollary. Let (CX , EX ) and (CY , EY ) be exact k-linear categories with enough projectives and

∗ f f∗ (CX , EX ) −−−→ (CY , EY ) −−−→ (CX , EX )

∗ a pair of ’exact’ functors such that f is k-linear and a left adjoint of f∗. Then the functor ∗ T−f ∗ f induces a triangle k-linear functor T−CS−X −−−→ T−CS−Y between the corresponding costable cosuspended categories. ∗ Proof. By K7.1, the functor f maps projective objects of (CX , EX ) to projective objects of (CY , EY ). The assertion follows now from K9.3.

K10. Complements.

K10.1. Admissible morphisms. Let (CX , EX ) be an exact k-linear category with the class of inﬂations MX and the class of deﬂations EX . We call arrows of MX ◦ EX admissible. In general, the class of admissible morphisms is not closed under composition.

j j0 K10.1.1. Lemma. Suppose that for any pair of arrows L −→ M ←− Le of MX , there exists a cartesian square j00 M −−−→ L f e   ej y y j (1) j0 L −−−→ M Then the class of admissible arrows is closed under composition. 0 Proof. (i) Notice that if (1) is a cartesian square with j ∈ MX 3 j , then the remaining 00 00 two arrows, j and ej, belong to MX too. In fact, the arrows j and ej are (strict) monomorphisms in any category. The Gabriel-Quillen embedding, preserves cartesian squares, maps arrows of MX to monomorphisms, and reﬂects monomorphisms to arrows of MX . j (ii) It suﬃces to show that EX ◦ MX ⊆ MX ◦ EX . Let L −→ M be a morphism of e MX and M −→ N a morphism of EX . Then we have a commutative diagram

j e 0 −−−→ Ker(e) −−−→e L −−−→e M 0 −−−→ 0 e   00    0 (2) j y j y y j je e 0 −−−→ Ker(e) −−−→ M −−−→ N −−−→ 0

194 with exact rows. Its left square is cartesian and formed by arrows of MX . The morphism e 0 L −→e M is a cokernel of ej; in particular, belongs to EX . The existence of the right 0 vertical arrow in (2), M 0 −→j N, follows from the exactness of the rows. Applying the ∗ Gabriel-Quillen embedding, jX . to the diagram (2), we reduce to the case of an abelian ∗ 0 category with the canonical exact structure. One can see that jX (j ) is a monomorphism. 0 0 0 Therefore, j is an arrow of MX . Thus, we obtain the equality e ◦ j = j ◦e, where j ∈ MX and e ∈ EX . K10.1.2. Remarks. (a) If the condition of K10.1.1 holds, then the dual condition 0 0 e e holds for deﬂations. In fact, let N ←− M −→ N be a pair of arrows of EX . So that we 0 0 have exact sequences 0 −→ L0 −→j M −→e N 0 −→ 0 and 0 −→ L −→j M −→e N −→ 0. By hypothesis (and the part (i) of the argument above), there is a cartesian square

j L −−−→e L0 e  00  0 j y y j j L −−−→ M

00 with all arrows from MX . Since j ◦ j ∈ MX , there is an exact sequence

00 j◦j e1 0 −−−→ Le −−−→ M −−−→ Ne −−−→ 0.

By the universal properties of cokernels, there exists a commutative square

e M −−−→ N   0   00 e y y e (3) e 0 e N −−−→ Ne

00 00 0 with arrows e and e uniquely determined by the equalities e ◦ e = e1 = e ◦ e . Since 00 e1 ∈ EX , it follows, by a property of exact categories, that e and e are arrows of EX . It is easy to see that the square (3) is cocartesian. (b) The assumption of K10.1.1 holds for exact categories associated with quasi-abelian categories (discussed shortly in K10.2 below), because in quasi-abelian categories all ﬁbred products and coproducts exist, MX is the class of all strict monomorphisms, and a pullback of a strict monomorphism is a strict monomorphism. K10.1.3. Proposition. Suppose the condition of K10.1.1 holds. Then the class of all admissible morphisms of the exact category (CX , EX ) forms the largest abelian exact subcategory, CXa(E), of (CX , EX ).

g −→ Proof. Let M −→ N be a pair of morphisms of CX . Their sum is the composition of h the arrows ∆M g⊕h +N M −−−→ M ⊕ M −−−→ N ⊕ N −−−→ N, (4)

195 where ∆M is the diagonal morphism and +N is the codiagonal morphism. Since the composition of ∆M and any of projections M ⊕M −→ M is the identical morphism, ∆M ∈ M.

Dually, +N belongs to EX . If both g and h are admissible arrows, then g ⊕ h is admissible. Therefore, in this case, g + h is the composition of admissible morphisms. Under the condition of K10.1.1, the composition of admissible morphisms is an admissible morphism.

The subcategory CXa(E) has same objects as CX . Therefore, since the category CX is additive, CXa(E) is additive too. It is quasi-abelian, because every admissible morphism has a kernel and a cokernel. An admissible arrow is a monomorphism iff it belongs to MX . Since all arrows of MX are strict monomorphisms, an inflation is an epimorphism iff it is an isomorphism. Altogether means that CXa(E) is an abelian subcategory. The exact structure EX induces the canonical exact structure on the subcategory CXa(E). It follows that any other abelian exact subcategory of (CX , EX ) is formed by admissible arrows, i.e. it is contained in CXa(E).

K10.1.4. Example: the category of torsion-free objects. Let (CX , EX ) be an 0 exact k-linear category. Let T be a full subcategory of CX such that if M −→ M is an inflation and M ∈ ObT , then M 0 is an object of T too. In particular, the subcategory T is strictly full. Let CXT denote the full subcategory of CX generated by all T -torsion free objects; i.e. objects N such that the only inflation L −→ N with L ∈ ObT is zero. K10.1.4.1. Lemma. Suppose that for any pair L0 −→ L ←− L00 of inflations of 0 00 (CX , EX ), there exists a pull-back L ×L L . Then the subcategory CXT of T -torsion free objects is closed under extensions. In particular, CXT is an exact subcategory of (CX , EX ).

0 j e 00 0 Proof. Let M −→ M −→ M be a conﬂation with M ∈ ObCXT . Let L −→ M be an inﬂation with L ∈ ObT . Then we have a commutative diagram

j0 e0 L0 −−−→ L −−−→ L00       y y y (1) j e M 0 −−−→ M −−−→ M 00 whose left square is cartesian and the both rows are conflations. In fact, by K10.1.2(a), all arrows of the left square are inflations. The arrow e0 is the cokernel of j0. It follows from the argument of K10.1.3 (or direct application of the Gabriel- Quillen embedding and the corresponding fact for abelian categories) that the remaining (right) vertical arrow is an inflation too. Since L0 ∈ ObT and M 0 is T -torsion free, it follows that L0 = 0 therefore e0 is an isomorphism. Therefore, if M 00 is also T -torsion free, then L00 = 0 which implies that L = 0. This shows that if the ends of a conflation are T -torsion free, same holds for the middle.

K10.2. Quasi-abelian categories. A quasi-abelian category is an additive category CX with kernels and cokernels and such that every pullback of a strict epimorphism is a strict epimorphism, and every pushout of a strict monomorphism is a strict monomorphism. It follows from deﬁnitions that the pair (CX , Es), where Es is the class of all short exact sequences in CX , is an exact category.

196 Every abelian category is quasi-abelian.

K10.2.1. Proposition. Let CX be a quasi-abelian category. There exist two canonical fully faithful functors CLX ←-CX ,→ CRX of CX into abelian categories which preserve and reﬂect exactness. Moreover, the category CX is stable under extensions in these embeddings. The category CX is closed under taking subobjects in CLX and every object of CLX is a quotient of an object of CX . Dually, CX is closed under taking quotients in CRX and every object of CRX is a subobject of an object of CX . Proof. See [Sch, 1.2.35, 1.2.31].

K10.2.2. Quasi-abelian categories and torsion pairs. Let CX be a quasi-abelian category, and let (T , F) be a torsion pair in CX . That is T and F are full subcategories ⊥ of CX such that F ⊆ T and CX = T•F. The latter means that every object M of CX ﬁts into an exact sequence

0 −→ M 0 −→j M −→e M 00 −→ 0 (1) with M 0 ∈ T and M 00 ∈ ObF. Notice that the exact sequence (1) is unique up to isomor- f phism. In fact, if N −→ M is a morphism and N ∈ ObT , then e ◦ f = 0, hence f factors uniquely through the monomorphism M 0 −→j M. This implies, in particular, that T is closed under taking quotients (in CX ) and, dually, F is closed under taking strict subobjects. 0 00 jT ∗ The assignments M 7−→ M and M 7−→ M in (1) extend to functors CX −→ T and ∗ jT ! jT CX −→ F which are resp. a right and a left adjoint to the inclusion functors T −→ CX ∗ jF and F −→ CX . By [GZ, 1], the categories T and F have all types of limits and colimits which exist in the category CX given by the formulas

∗ ∗ lim D = jT ∗(lim(jT ◦ D)) and colimD = jT ∗(colim(jT ◦ D)) (2) for any small diagram D −→D T . In particular, T has kernels and cokernels given by

CokerT = CokerCX and KerT = jT ∗(KerT ). Similarly for F.

A torsion pair (T , F) in CX is called tilting if every object of CX is a subobject of an object of T . Dually, (T , F) is called a cotilting torsion pair if every object of CX is a quotient of an object of F.

K10.2.2.1. Proposition. Let CX be an additive category. The following conditions are equivalent. (a) CX is quasi-abelian. (b) There exists a tilting torsion pair (T , F) in an abelian category CY such that T is equivalent to CX . 0 0 (c) There exists a cotilting torsion pair (T , F ) in an abelian category CW such that 0 F is equivalent to CX .

Proof. It follows from K10.2.1 that CY = CRX and CY = CLX . See details of the proof in [BOVdB, B.3].

197 References [Ba] H. Bass, Algebraic K-theory, Benjamin 1968 [BOVdB] A. Bondal, M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, math.AG/0204218 v2 [CE] H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, 1956 [Gab] P. Gabriel, Des catégoriesabéliennes,Bull. Soc. Math. France, 90 (1962), 323-449 [GZ] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Springer Verlag, Berlin-Heidelberg-New York, 1967 [Gr] A. Grothendieck, Sur quelques points d’algèbre homologique. Tohuku Math. J., Ser. II, 9, 120–221 (1957) [GrD] A. Grothendieck, J.A. Dieudonné,Eléments de GéométrieAlgébrique,Springer Verlag, New York - Heidelberg - Berlin, 1971 [Ke1] B. Keller, Derived categories and their uses, preprint, 1994. [Ke2] B. Keller, Chain complexes and stable categories, Manus. Math. 67 (1990), 379–417. [KeV] B. Keller, D. Vossieck, Sous les catégoriesdérivées, C. R. Acad. Sci. Paris 305 (1987), 225–228. [KeV1] B. Keller, D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Sér.A 40 (1988), no. 2, 239–253. [KR] M. Kontsevich, A.L. Rosenberg, Noncommutative spaces and flat descent, preprint MPIM, 2004(36), 108 pp. [ML] S. Mac-Lane, Categories for the working mathematicians, Springer - Verlag; New York - Heidelberg - Berlin (1971) [MLM] S. Mac-Lane, L. Moerdijk, Sheaves in Geometry and Logic, Springer - Verlag; New York - Heidelberg - Berlin (1992) [Q] D. Quillen, Higher algebraic K-theory, LNM 341, 85–147 (1973) [R] A.L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, Kluwer Academic Publishers, Mathematics and Its Applications, v.330 (1995), 328 pages. [R1] A.L. Rosenberg, Noncommutative spaces and schemes, preprint MPIM, 1999 [R2] A.L. Rosenberg, Trieste Lectures, to appear in Lecture Notes of the School on Alge- braic K-Theory and Applications, ICTP, Italy, 92 pp. [R3] A.L. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM, 2008(57), 105 pp [R4] A.L. Rosenberg, Geometry of right exact ’spaces’, preprint MPIM, 2008(85), 66 pp. [Ve1] J.–L. Verdier, Catégoriesdérivées, Séminairede GéométrieAlgébrique4 1/2, Coho- mologie Etale,´ LNM 569, pp. 262–311, Springer, 1977 [Ve2] J.–L. Verdier, Des catégoriesdérivéesdes catégoriesabéliennes,Astérisque,v.239, 1996

198